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Vector bundle

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a vector bundle is a <a href="Topological" class="mw-redirect" title="Topological">topological</a> construction that makes precise the idea of a <a href="Family_of_sets" title="Family of sets">family</a> of <a href="Vector_space" title="Vector space">vector spaces</a> parameterized by another <a href="Space_(mathematics)" title="Space (mathematics)">space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (for example <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> could be a <a href="Topological_space" title="Topological space">topological space</a>, a <a href="Manifold" title="Manifold">manifold</a>, or an <a href="Algebraic_variety" title="Algebraic variety">algebraic variety</a>): to every point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> of the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> we associate (or "attach") a vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)}">

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<annotation encoding="application/x-tex">{\displaystyle V(x)}</annotation>

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</math><img src="./4ab3e825c2bf9c80d11d12e070a4626d48e03c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.926ex; height:2.843ex;" alt="{\displaystyle V(x)}" loading="lazy"> in such a way that these vector spaces fit together to form another space of the same kind as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">.

The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)=V}">

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<annotation encoding="application/x-tex">{\displaystyle V(x)=V}</annotation>

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</math><img src="./e7c3b02ea27a0711748cda32ad3ddb07ecfe7883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.812ex; height:2.843ex;" alt="{\displaystyle V(x)=V}" loading="lazy"> <a href="For_all" class="mw-redirect" title="For all">for all</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">: in this case there is a copy of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and these copies fit together to form the vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times V}">

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<annotation encoding="application/x-tex">{\displaystyle X\times V}</annotation>

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</math><img src="./7debf43b00ac69f7b4dbaacb835837ff4e63d358.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.608ex; height:2.176ex;" alt="{\displaystyle X\times V}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. Such vector bundles are said to be <a href="Fiber_bundle#Trivial_bundle" title="Fiber bundle">trivial</a>. A more complicated (and prototypical) class of examples are the <a href="Tangent_bundle" title="Tangent bundle">tangent bundles</a> of <a href="Manifold" title="Manifold">smooth (or differentiable) manifolds</a>: to every point of such a manifold we attach the <a href="Tangent_space" title="Tangent space">tangent space</a> to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the <a href="Hairy_ball_theorem" title="Hairy ball theorem">hairy ball theorem</a>. In general, a manifold is said to be <a href="Parallelizable_manifold" title="Parallelizable manifold">parallelizable</a> if, and only if, its tangent bundle is trivial.

Vector bundles are almost always required to be locally trivial, which means they are examples of <a href="Fiber_bundle" title="Fiber bundle">fiber bundles</a>. Also, the vector spaces are usually required to be over the <a href="Real_number" title="Real number">real</a> or <a href="Complex_number" title="Complex number">complex numbers</a>, in which case the vector bundle is said to be a real or complex vector bundle (respectively). <a href="Complex_vector_bundle" title="Complex vector bundle">Complex vector bundles</a> can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the <a href="Category_of_topological_spaces" title="Category of topological spaces">category of topological spaces</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition_and_first_consequences">Definition and first consequences</h2></div>

A real vector bundle consists of:

<ol><li>topological spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (base space) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

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</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> (total space)</li>

<li>a <a href="Continuous_function" title="Continuous function">continuous</a> <a href="Surjection" class="mw-redirect" title="Surjection">surjection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :E\to X}">

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<annotation encoding="application/x-tex">{\displaystyle \pi :E\to X}</annotation>

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</math><img src="./ebbf8de3c25d0905f50abaf4f9374daa0a4bd796.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.639ex; height:2.176ex;" alt="{\displaystyle \pi :E\to X}" loading="lazy"> (bundle projection)</li>

<li>for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, the structure of a <a href="Hamel_dimension" class="mw-redirect" title="Hamel dimension">finite-dimensional</a> <a href="Real_number" title="Real number">real</a> <a href="Vector_space" title="Vector space">vector space</a> on the <a href="Fiber_bundle" title="Fiber bundle">fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-1}(\{x\})}">

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<annotation encoding="application/x-tex">{\displaystyle \pi ^{-1}(\{x\})}</annotation>

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</math><img src="./1324bc8d82b0ad4271b2ea25dcd89f6fcf936a56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.131ex; height:3.176ex;" alt="{\displaystyle \pi ^{-1}(\{x\})}" loading="lazy"></li></ol>

where the following compatibility condition is satisfied: for every point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, there is an <a href="Open_neighborhood" class="mw-redirect" title="Open neighborhood">open neighborhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq X}">

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<annotation encoding="application/x-tex">{\displaystyle U\subseteq X}</annotation>

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</math><img src="./bc4b6027fd83ac32e7b4f4113c60a69041292723.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.861ex; height:2.343ex;" alt="{\displaystyle U\subseteq X}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, a <a href="Natural_number" title="Natural number">natural number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, and a <a href="Homeomorphism" title="Homeomorphism">homeomorphism</a>

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<annotation encoding="application/x-tex">{\displaystyle \varphi \colon U\times \mathbb {R} ^{k}\to \pi ^{-1}(U)}</annotation>

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</math><img src="./78126d4f5262fa66874845674c8fe06cf75ee243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.817ex; height:3.176ex;" alt="{\displaystyle \varphi \colon U\times \mathbb {R} ^{k}\to \pi ^{-1}(U)}" loading="lazy"></dd></dl>

such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">,

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<annotation encoding="application/x-tex">{\displaystyle (\pi \circ \varphi )(x,v)=x}</annotation>

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</math><img src="./9941764a8a38d21116503cfdf3226dc1d87884d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.585ex; height:2.843ex;" alt="{\displaystyle (\pi \circ \varphi )(x,v)=x}" loading="lazy"> for all <a href="Euclidean_vector" title="Euclidean vector">vectors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{k}}</annotation>

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</math><img src="./1bcd8908c9fa46eb979ef7b67d1bb65eb3692cbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{k}}" loading="lazy">, and</li>

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<annotation encoding="application/x-tex">{\displaystyle v\mapsto \varphi (x,v)}</annotation>

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</math><img src="./c91df1d568d242f410e1136d954b578668790221.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.562ex; height:2.843ex;" alt="{\displaystyle v\mapsto \varphi (x,v)}" loading="lazy"> is a <a href="Linear_map" title="Linear map">linear</a> <a href="Isomorphism" title="Isomorphism">isomorphism</a> between the vector spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{k}}</annotation>

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</math><img src="./1bcd8908c9fa46eb979ef7b67d1bb65eb3692cbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{k}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-1}(\{x\})}">

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<annotation encoding="application/x-tex">{\displaystyle \pi ^{-1}(\{x\})}</annotation>

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</math><img src="./1324bc8d82b0ad4271b2ea25dcd89f6fcf936a56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.131ex; height:3.176ex;" alt="{\displaystyle \pi ^{-1}(\{x\})}" loading="lazy">.</li></ul>

The open neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> together with the homeomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is called a <a href="Local_trivialization" class="mw-redirect" title="Local trivialization">local trivialization</a> of the vector bundle. The local trivialization shows that locally the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> "looks like" the projection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\times \mathbb {R} ^{k}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle U\times \mathbb {R} ^{k}}</annotation>

</semantics>

</math><img src="./6b80843ab74234e43f73e372586de584aade0bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.39ex; height:2.676ex;" alt="{\displaystyle U\times \mathbb {R} ^{k}}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">.

Every fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-1}(\{x\})}">

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle \pi ^{-1}(\{x\})}</annotation>

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</math><img src="./1324bc8d82b0ad4271b2ea25dcd89f6fcf936a56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.131ex; height:3.176ex;" alt="{\displaystyle \pi ^{-1}(\{x\})}" loading="lazy"> is a finite-dimensional real vector space and hence has a <a href="Dimension" title="Dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{x}}</annotation>

</semantics>

</math><img src="./adcbcc5a960cbbd063fb012b087a8fdef03066d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.509ex;" alt="{\displaystyle k_{x}}" loading="lazy">. The local trivializations show that the <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to k_{x}}">

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle x\to k_{x}}</annotation>

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</math><img src="./44202a2dfa93a48f09d8026cf6a8c79d1de7701c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.327ex; height:2.509ex;" alt="{\displaystyle x\to k_{x}}" loading="lazy"> is <a href="Locally_constant" class="mw-redirect" title="Locally constant">locally constant</a>, and is therefore constant on each <a href="Locally_connected_space" title="Locally connected space">connected component</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{x}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle k_{x}}</annotation>

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</math><img src="./adcbcc5a960cbbd063fb012b087a8fdef03066d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.509ex;" alt="{\displaystyle k_{x}}" loading="lazy"> is equal to a constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> on all of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is called the rank of the vector bundle, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> is said to be a vector bundle of rank <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. Often the definition of a vector bundle includes that the rank is well defined, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{x}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle k_{x}}</annotation>

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</math><img src="./adcbcc5a960cbbd063fb012b087a8fdef03066d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.509ex;" alt="{\displaystyle k_{x}}" loading="lazy"> is constant. Vector bundles of rank 1 are called <a href="Line_bundle" title="Line bundle">line bundles</a>, while those of rank 2 are less commonly called plane bundles.

The <a href="Cartesian_product" title="Cartesian product">Cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times \mathbb {R} ^{k}}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle X\times \mathbb {R} ^{k}}</annotation>

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</math><img src="./8bb393f0441bbee93188d64d08ec6710993bd2f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.587ex; height:2.676ex;" alt="{\displaystyle X\times \mathbb {R} ^{k}}" loading="lazy">, equipped with the projection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times \mathbb {R} ^{k}\to X}">

<msup>

</mrow>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times \mathbb {R} ^{k}\to X}</annotation>

</semantics>

</math><img src="./01c022dc9321d0c05ef498784bf67de967698cbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.181ex; height:2.676ex;" alt="{\displaystyle X\times \mathbb {R} ^{k}\to X}" loading="lazy">, is called the trivial bundle of rank <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Transition_functions">Transition functions</h3></div>

Given a vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\to X}</annotation>

</semantics>

</math><img src="./c6e385963fc9a729e22866cd3c05d5002c361bbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.37ex; height:2.176ex;" alt="{\displaystyle E\to X}" loading="lazy"> of rank <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, and a pair of neighborhoods <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> over which the bundle trivializes via

<mtr>

<mtd>

<msub>

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</msub>

<msup>

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</msup>

</mtd>

<mtd>

<mover>

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</mover>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

<mover>

</mpadded>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varphi _{U}\colon U\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(U),\\\varphi _{V}\colon V\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(V)\end{aligned}}}</annotation>

</semantics>

</math><img src="./83d6925f8dbabbb8fa8d3b0c26d951c4c61aaf40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; margin-top: -0.339ex; width:23.994ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}\varphi _{U}\colon U\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(U),\\\varphi _{V}\colon V\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(V)\end{aligned}}}" loading="lazy"></dd></dl>

the <a href="Composite_Function" class="mw-redirect" title="Composite Function">composite function</a>

</mrow>

</mrow>

</msubsup>

<msub>

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</msub>

<msup>

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</mrow>

</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{U}^{-1}\circ \varphi _{V}\colon (U\cap V)\times \mathbb {R} ^{k}\to (U\cap V)\times \mathbb {R} ^{k}}</annotation>

</semantics>

</math><img src="./3d4194d4ebcfe431f84d3377d9705429b467c63a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.85ex; height:3.343ex;" alt="{\displaystyle \varphi _{U}^{-1}\circ \varphi _{V}\colon (U\cap V)\times \mathbb {R} ^{k}\to (U\cap V)\times \mathbb {R} ^{k}}" loading="lazy"></dd></dl>

is well-defined on the overlap, and satisfies

</mrow>

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</msubsup>

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{U}^{-1}\circ \varphi _{V}(x,v)=\left(x,g_{UV}(x)v\right)}</annotation>

</semantics>

</math><img src="./6aeaf146c9756b97ddb6867046214233491a7127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.768ex; height:3.343ex;" alt="{\displaystyle \varphi _{U}^{-1}\circ \varphi _{V}(x,v)=\left(x,g_{UV}(x)v\right)}" loading="lazy"></dd></dl>

for some <a href="General_linear_group" title="General linear group"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{GL}}(k)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{GL}}(k)}</annotation>

</semantics>

</math><img src="./4fab8ddbf7d0cd2b831e7801dd08b8e15039b2da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.298ex; height:2.843ex;" alt="{\displaystyle {\text{GL}}(k)}" loading="lazy">-valued</a> function

<msub>

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<annotation encoding="application/x-tex">{\displaystyle g_{UV}\colon U\cap V\to \operatorname {GL} (k).}</annotation>

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</math><img src="./2cb009b693db02865b497fc99a228e5a832ed1e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.611ex; height:2.843ex;" alt="{\displaystyle g_{UV}\colon U\cap V\to \operatorname {GL} (k).}" loading="lazy"></dd></dl>

These are called the <a href="Transition_map" class="mw-redirect" title="Transition map">transition functions</a> (or the coordinate transformations) of the vector bundle.

The <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of transition functions forms a <a href="%C4%8Cech_cocycle" class="mw-redirect" title="Čech cocycle">Čech cocycle</a> in the sense that

<msub>

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</msub>

<msub>

</mrow>

</msub>

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle g_{UU}(x)=I,\quad g_{UV}(x)g_{VW}(x)g_{WU}(x)=I}</annotation>

</semantics>

</math><img src="./9354e02750aa109d0a9476cdfd75636134050902.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.831ex; height:2.843ex;" alt="{\displaystyle g_{UU}(x)=I,\quad g_{UV}(x)g_{VW}(x)g_{WU}(x)=I}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U,V,W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U,V,W}</annotation>

</semantics>

</math><img src="./9f4d31f98b67bd24af02cbb34ade648ad59dd201.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.073ex; height:2.509ex;" alt="{\displaystyle U,V,W}" loading="lazy"> over which the bundle trivializes satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cap V\cap W\neq \emptyset }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\cap V\cap W\neq \emptyset }</annotation>

</semantics>

</math><img src="./0dcf743f4101ee66a1bbd8a44e762e3d95d52d3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.431ex; height:2.843ex;" alt="{\displaystyle U\cap V\cap W\neq \emptyset }" loading="lazy">. Thus the data <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}</annotation>

</semantics>

</math><img src="./7d522e4b89a83e991a5f3c4d3d9068b0488280ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.766ex; height:3.176ex;" alt="{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}" loading="lazy"> defines a <a href="Fiber_bundle" title="Fiber bundle">fiber bundle</a>; the additional data of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{UV}}">

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle g_{UV}}</annotation>

</semantics>

</math><img src="./eb934498b382f3276942bdd431e1d435a9cdb76f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.866ex; height:2.009ex;" alt="{\displaystyle g_{UV}}" loading="lazy"> specifies a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{GL}}(k)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{GL}}(k)}</annotation>

</semantics>

</math><img src="./4fab8ddbf7d0cd2b831e7801dd08b8e15039b2da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.298ex; height:2.843ex;" alt="{\displaystyle {\text{GL}}(k)}" loading="lazy"> structure group in which the <a href="Group_action" title="Group action">action</a> on the fiber is the standard action of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{GL}}(k)}">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\text{GL}}(k)}</annotation>

</semantics>

</math><img src="./4fab8ddbf7d0cd2b831e7801dd08b8e15039b2da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.298ex; height:2.843ex;" alt="{\displaystyle {\text{GL}}(k)}" loading="lazy">.

Conversely, given a fiber bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}</annotation>

</semantics>

</math><img src="./7d522e4b89a83e991a5f3c4d3d9068b0488280ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.766ex; height:3.176ex;" alt="{\displaystyle (E,X,\pi ,\mathbb {R} ^{k})}" loading="lazy"> with a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{GL}}(k)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{GL}}(k)}</annotation>

</semantics>

</math><img src="./4fab8ddbf7d0cd2b831e7801dd08b8e15039b2da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.298ex; height:2.843ex;" alt="{\displaystyle {\text{GL}}(k)}" loading="lazy"> cocycle acting in the standard way on the fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{k}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{k}}</annotation>

</semantics>

</math><img src="./1bcd8908c9fa46eb979ef7b67d1bb65eb3692cbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{k}}" loading="lazy">, there is <a href="Associated_bundle" title="Associated bundle">associated</a> a vector bundle. This is an example of the <a href="Fibre_bundle_construction_theorem" class="mw-redirect" title="Fibre bundle construction theorem">fibre bundle construction theorem</a> for vector bundles, and can be taken as an alternative definition of a vector bundle.

<div class="mw-heading mw-heading3"><h3 id="Subbundles">Subbundles</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Subbundle" title="Subbundle">Subbundle</a></div>

One simple method of constructing vector bundles is by taking subbundles of other vector bundles. Given a vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :E\to X}">

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<annotation encoding="application/x-tex">{\displaystyle \pi :E\to X}</annotation>

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</math><img src="./ebbf8de3c25d0905f50abaf4f9374daa0a4bd796.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.639ex; height:2.176ex;" alt="{\displaystyle \pi :E\to X}" loading="lazy"> over a topological space, a subbundle is simply a <a href="Topological_subspace" class="mw-redirect" title="Topological subspace">topological subspace</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\subset E}">

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<annotation encoding="application/x-tex">{\displaystyle F\subset E}</annotation>

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</math><img src="./2dc790c91db34007686074e392dfed5bb12f482e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.615ex; height:2.176ex;" alt="{\displaystyle F\subset E}" loading="lazy"> for which the <a href="Restriction_of_a_map" class="mw-redirect" title="Restriction of a map">restriction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.\pi \right|_{F}}">

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<annotation encoding="application/x-tex">{\displaystyle \left.\pi \right|_{F}}</annotation>

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</math><img src="./4d446dfad36ee9b74b7494280da783f9dfb2791b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.442ex; height:2.843ex;" alt="{\displaystyle \left.\pi \right|_{F}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.\pi \right|_{F}:F\to X}">

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<annotation encoding="application/x-tex">{\displaystyle \left.\pi \right|_{F}:F\to X}</annotation>

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</math><img src="./04af20074643e5c0c4cad037cf69a20f8c76de72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.714ex; height:2.843ex;" alt="{\displaystyle \left.\pi \right|_{F}:F\to X}" loading="lazy"> the structure of a vector bundle also. In this case the fibre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{x}\subset E_{x}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{x}\subset E_{x}}</annotation>

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</math><img src="./00bce5646d7964a1281ed33a9c75a413183a278f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.653ex; height:2.509ex;" alt="{\displaystyle F_{x}\subset E_{x}}" loading="lazy"> is a vector subspace for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

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</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy">.

A subbundle of a trivial bundle need not be trivial, and indeed every real vector bundle over a compact space can be viewed as a subbundle of a trivial bundle of sufficiently high rank. For example, the <a href="M%C3%B6bius_band" class="mw-redirect" title="Möbius band">Möbius band</a>, a non-trivial <a href="Line_bundle" title="Line bundle">line bundle</a> over the circle, can be seen as a subbundle of the trivial rank 2 bundle over the circle.

<div class="mw-heading mw-heading2"><h2 id="Vector_bundle_morphisms">Vector bundle morphisms</h2></div>

A <a href="Morphism" title="Morphism">morphism</a> from the vector bundle π1: E1 → X1 to the vector bundle π2: E2 → X2 is given by a pair of continuous maps f: E1 → E2 and g: X1 → X2 such that

<dl><dd>g ∘ π1 = π2 ∘ f

<dd>for every x in X1, the map π1−1({x}) → π2−1({g(x)}) <a href="Induced_map" class="mw-redirect" title="Induced map">induced</a> by f is a <a href="Linear_map" title="Linear map">linear map</a> between vector spaces.</dd></dl>

Note that g is determined by f (because π1 is surjective), and f is then said to cover g.

The class of all vector bundles together with bundle morphisms forms a <a href="Category_(mathematics)" title="Category (mathematics)">category</a>. Restricting to vector bundles for which the spaces are manifolds (and the bundle projections are smooth maps) and smooth bundle morphisms we obtain the category of smooth vector bundles. Vector bundle morphisms are a special case of the notion of a <a href="Bundle_map" title="Bundle map">bundle map</a> between <a href="Fiber_bundle" title="Fiber bundle">fiber bundles</a>, and are sometimes called (vector) bundle homomorphisms.

A bundle homomorphism from E1 to E2 with an <a href="Inverse_element" title="Inverse element">inverse</a> which is also a bundle homomorphism (from E2 to E1) is called a (vector) bundle isomorphism, and then E1 and E2 are said to be isomorphic vector bundles. An isomorphism of a (rank k) vector bundle E over X with the trivial bundle (of rank k over X) is called a trivialization of E, and E is then said to be trivial (or trivializable). The definition of a vector bundle shows that any vector bundle is locally trivial.

We can also consider the category of all vector bundles over a fixed base space X. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the <a href="Identity_function" title="Identity function">identity map</a> on X. That is, bundle morphisms for which the following diagram <a href="Commutative_diagram" title="Commutative diagram">commutes</a>:

(Note that this category is not <a href="Abelian_category" title="Abelian category">abelian</a>; the <a href="Kernel_(category_theory)" title="Kernel (category theory)">kernel</a> of a morphism of vector bundles is in general not a vector bundle in any natural way.)

A vector bundle morphism between vector bundles π1: E1 → X1 and π2: E2 → X2 covering a map g from X1 to X2 can also be viewed as a vector bundle morphism over X1 from E1 to the <a href="Pullback_bundle" title="Pullback bundle">pullback bundle</a> g*E2.

<div class="mw-heading mw-heading2"><h2 id="Sections_and_locally_free_sheaves">Sections and locally free sheaves</h2></div>

Given a vector bundle π: E → X and an open subset U of X, we can consider <a href="Section_(fiber_bundle)" title="Section (fiber bundle)">sections</a> of π on U, i.e. continuous functions s: U → E where the composite π ∘ s is such that (π ∘ s)(u) = u for all u in U. A section over U is an assignment, to every point p of U, a vector from the vector space fibre above p, in a continuous manner. As an example, a section of the tangent bundle of a differential manifold is the same as a <a href="Vector_field" title="Vector field">vector field</a> on that manifold.

Let F(U) be the set of all sections on U. F(U) always contains at least one element, namely the zero section: the function s that maps every element x of U to the <a href="Zero_vector" class="mw-redirect" title="Zero vector">zero element of the vector space</a> π−1({x}). With the <a href="Pointwise" title="Pointwise">pointwise</a> addition and <a href="Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> of sections, F(U) becomes itself a real vector space. The collection of these vector spaces is a <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> of vector spaces on X.

If s is an element of F(U) and f : U → R is a continuous map, then their product fs (pointwise scalar multiplication) is in F(U). This shows that F(U) is a <a href="Module_(mathematics)" title="Module (mathematics)">module</a> over the <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> of continuous <a href="Real-valued_function" title="Real-valued function">real-valued functions</a> on U. Furthermore, if OX denotes the structure sheaf of continuous real-valued functions on X, then F becomes a sheaf of OX-modules.

Not every sheaf of OX-modules arises in this fashion from a vector bundle: only the <a href="Locally_free_sheaf" class="mw-redirect" title="Locally free sheaf">locally free</a> ones do. (The reason is that locally we are looking for sections of a projection U × Rk → U; these are precisely the continuous functions U → Rk, and such a function is a k-<a href="Tuple" title="Tuple">tuple</a> of continuous functions U → R.)

Even more: the category of real vector bundles on X is <a href="Equivalence_of_categories" title="Equivalence of categories">equivalent</a> to the category of locally free and <a href="Finitely_generated_module" title="Finitely generated module">finitely generated</a> sheaves of OX-modules.

So we can think of the category of real vector bundles on X as sitting inside the category of <a href="Sheaf_of_modules" title="Sheaf of modules">sheaves of OX-modules</a>; this latter category is abelian, so this is where we can compute <a href="Kernel_(algebra)" title="Kernel (algebra)">kernels</a> and <a href="Cokernel" title="Cokernel">cokernels</a> of morphisms of vector bundles.

A rank n vector bundle is trivial <a href="If_and_only_if" title="If and only if">if and only if</a> it has n <a href="Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> global sections.

<div class="mw-heading mw-heading2"><h2 id="Operations_on_vector_bundles">Operations on vector bundles</h2></div>

Most <a href="Operation_(mathematics)" title="Operation (mathematics)">operations</a> on vector spaces can be extended to vector bundles by performing the vector space operation fiberwise.

For example, if E is a vector bundle over X, then there is a bundle E* over X, called the <a href="Dual_bundle" title="Dual bundle">dual bundle</a>, whose fiber at x ∈ X is the <a href="Dual_vector_space" class="mw-redirect" title="Dual vector space">dual vector space</a> (Ex)*. Formally E* can be defined as the set of pairs (x, φ), where x ∈ X and φ ∈ (Ex)*. The dual bundle is locally trivial because the <a href="Transpose" title="Transpose">dual space</a> of the inverse of a local trivialization of E is a local trivialization of E*: the key point here is that the operation of taking the dual vector space is <a href="Functorial" class="mw-redirect" title="Functorial">functorial</a>.

There are many functorial operations which can be performed on pairs of vector spaces (over the same field), and these extend straightforwardly to pairs of vector bundles E, F on X (over the given field). A few examples follow.

<ul><li>The Whitney sum (named for <a href="Hassler_Whitney" title="Hassler Whitney">Hassler Whitney</a>) or direct sum bundle of E and F is a vector bundle E ⊕ F over X whose fiber over x is the <a href="Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> Ex ⊕ Fx of the vector spaces Ex and Fx.</li>

<li>The <a href="Tensor_product_bundle" title="Tensor product bundle">tensor product bundle</a> E ⊗ F is defined in a similar way, using fiberwise <a href="Tensor_product" title="Tensor product">tensor product</a> of vector spaces.</li>

<li>The Hom-bundle Hom(E, F) is a vector bundle whose fiber at x is the space of linear maps from Ex to Fx (which is often denoted Hom(Ex, Fx) or L(Ex, Fx)). The Hom-bundle is so-called (and useful) because there is a <a href="Bijection" title="Bijection">bijection</a> between vector bundle homomorphisms from E to F over X and sections of Hom(E, F) over X.</li>

<li>Building on the previous example, given a section s of an <a href="Endomorphism" title="Endomorphism">endomorphism</a> bundle Hom(E, E) and a function f: X → R, one can construct an eigenbundle by taking the fiber over a point x ∈ X to be the f(x)-<a href="Eigenvector" class="mw-redirect" title="Eigenvector">eigenspace</a> of the linear map s(x): Ex → Ex. Though this construction is natural, unless care is taken, the resulting object will not have local trivializations. Consider the case of s being the zero section and f having isolated zeroes. The fiber over these zeroes in the resulting "eigenbundle" will be isomorphic to the fiber over them in E, while everywhere else the fiber is the trivial 0-dimensional vector space.</li>

<li>The <a href="Dual_bundle" title="Dual bundle">dual vector bundle</a> E* is the Hom bundle Hom(E, R × X) of bundle homomorphisms of E and the trivial bundle R × X. There is a canonical vector bundle isomorphism Hom(E, F) = E* ⊗ F.</li></ul>

Each of these operations is a particular example of a general feature of bundles: that many operations that can be performed on the <a href="Category_of_vector_spaces" class="mw-redirect" title="Category of vector spaces">category of vector spaces</a> can also be performed on the category of vector bundles in a <a href="Functor" title="Functor">functorial</a> manner. This is made precise in the language of <a href="Smooth_functor" title="Smooth functor">smooth functors</a>. An operation of a different nature is the <a href="Pullback_bundle" title="Pullback bundle">pullback bundle</a> construction. Given a vector bundle E → Y and a continuous map f: X → Y one can "pull back" E to a vector bundle f*E over X. The fiber over a point x ∈ X is essentially just the fiber over f(x) ∈ Y. Hence, Whitney summing E ⊕ F can be defined as the pullback bundle of the diagonal map from X to X × X where the bundle over X × X is E × F.

Remark: Let X be a <a href="Compact_space" title="Compact space">compact space</a>. Any vector bundle E over X is a direct summand of a trivial bundle; i.e., there exists a bundle E' such that E ⊕ E' is trivial. This fails if X is not compact: for example, the <a href="Tautological_line_bundle" class="mw-redirect" title="Tautological line bundle">tautological line bundle</a> over the infinite real projective space does not have this property.<a href="#cite_note-FOOTNOTEHatcher2003Example_3.6-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Additional_structures_and_generalizations">Additional structures and generalizations</h2></div>

Vector bundles are often given more structure. For instance, vector bundles may be equipped with a <a href="Metric_(vector_bundle)" class="mw-redirect" title="Metric (vector bundle)">vector bundle metric</a>. Usually this metric is required to be <a href="Definite_bilinear_form" class="mw-redirect" title="Definite bilinear form">positive definite</a>, in which case each fibre of E becomes a <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>. A vector bundle with a <a href="Linear_complex_structure" title="Linear complex structure">complex structure</a> corresponds to a <a href="Complex_vector_bundle" title="Complex vector bundle">complex vector bundle</a>, which may also be obtained by replacing real vector spaces in the definition with complex ones and requiring that all mappings be <a href="Complex_linear_structure" class="mw-redirect" title="Complex linear structure">complex-linear</a> in the fibers. More generally, one can typically understand the additional structure imposed on a vector bundle in terms of the resulting <a href="Reduction_of_the_structure_group_of_a_bundle" class="mw-redirect" title="Reduction of the structure group of a bundle">reduction of the structure group of a bundle</a>. Vector bundles over more general <a href="Topological_field" class="mw-redirect" title="Topological field">topological fields</a> may also be used.

If instead of a finite-dimensional vector space, the fiber F is taken to be a <a href="Banach_space" title="Banach space">Banach space</a> then a <a href="Banach_bundle" title="Banach bundle">Banach bundle</a> is obtained.<a href="#cite_note-FOOTNOTELang1995-2">[2]</a> Specifically, one must require that the local trivializations are Banach space isomorphisms (rather than just linear isomorphisms) on each of the fibers and that, furthermore, the transitions

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<annotation encoding="application/x-tex">{\displaystyle g_{UV}\colon U\cap V\to \operatorname {GL} (F)}</annotation>

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</math><img src="./c4ee22af2ae10ffc1714836c5042461aa3f68b50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.493ex; height:2.843ex;" alt="{\displaystyle g_{UV}\colon U\cap V\to \operatorname {GL} (F)}" loading="lazy"></dd></dl>

are continuous mappings of <a href="Banach_manifold" title="Banach manifold">Banach manifolds</a>. In the corresponding theory for Cp bundles, all mappings are required to be Cp.

Vector bundles are special <a href="Fiber_bundle" title="Fiber bundle">fiber bundles</a>, those whose fibers are vector spaces and whose cocycle respects the vector space structure. More general fiber bundles can be constructed in which the fiber may have other structures; for example <a href="Sphere_bundle" title="Sphere bundle">sphere bundles</a> are fibered by spheres.

<div class="mw-heading mw-heading2"><h2 id="Smooth_vector_bundles">Smooth vector bundles</h2></div>

A vector bundle (E, p, M) is smooth, if E and M are <a href="Manifold" title="Manifold">smooth manifolds</a>, p: E → M is a smooth map, and the local trivializations are <a href="Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a>. Depending on the required degree of <a href="Smoothness" title="Smoothness">smoothness</a>, there are different corresponding notions of <a href="Continuously_differentiable" class="mw-redirect" title="Continuously differentiable">Cp</a> bundles, <a href="Infinitely_differentiable" class="mw-redirect" title="Infinitely differentiable">infinitely differentiable</a> C∞-bundles and <a href="Real_analytic" class="mw-redirect" title="Real analytic">real analytic</a> Cω-bundles. In this section we will concentrate on <a href="C-infinity" class="mw-redirect" title="C-infinity">C∞</a>-bundles. The most important example of a C∞-vector bundle is the <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> (TM, πTM, M) of a C∞-manifold M.

A smooth vector bundle can be characterized by the fact that it admits transition functions as described above which are smooth functions on overlaps of trivializing charts U and V. That is, a vector bundle E is smooth if it admits a covering by trivializing open sets such that for any two such sets U and V, the transition function

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<annotation encoding="application/x-tex">{\displaystyle g_{UV}:U\cap V\to \operatorname {GL} (k,\mathbb {R} )}</annotation>

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is a smooth function into the <a href="Matrix_group" class="mw-redirect" title="Matrix group">matrix group</a> GL(k,R), which is a <a href="Lie_group" title="Lie group">Lie group</a>.

Similarly, if the transition functions are:

<ul><li>Cr then the vector bundle is a Cr vector bundle,</li>

<li>real <a href="Analytic_function" title="Analytic function">analytic</a> then the vector bundle is a real analytic vector bundle (this requires the matrix group to have a real analytic structure),</li>

<li>holomorphic then the vector bundle is a <a href="Holomorphic_vector_bundle" title="Holomorphic vector bundle">holomorphic vector bundle</a> (this requires the matrix group to be a <a href="Complex_Lie_group" title="Complex Lie group">complex Lie group</a>),</li>

<li>algebraic functions then the vector bundle is an <a href="Algebraic_vector_bundle" class="mw-redirect" title="Algebraic vector bundle">algebraic vector bundle</a> (this requires the matrix group to be an <a href="Algebraic_group" title="Algebraic group">algebraic group</a>).</li></ul>

The C∞-vector bundles (E, p, M) have a very important property not shared by more general C∞-fibre bundles. Namely, the tangent space Tv(Ex) at any v ∈ Ex can be naturally identified with the fibre Ex itself. This identification is obtained through the vertical lift vlv: Ex → Tv(Ex), defined as

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {vl} _{v}w[f]:=\left.{\frac {d}{dt}}\right|_{t=0}f(v+tw),\quad f\in C^{\infty }(E_{x}).}</annotation>

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</math><img src="./afd374a28264c91b99a13a8fb0313d29e97537e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.79ex; height:5.843ex;" alt="{\displaystyle \operatorname {vl} _{v}w[f]:=\left.{\frac {d}{dt}}\right|_{t=0}f(v+tw),\quad f\in C^{\infty }(E_{x}).}" loading="lazy"></dd></dl>

The vertical lift can also be seen as a natural C∞-vector bundle isomorphism p*E → VE, where (p*E, p*p, E) is the pull-back bundle of (E, p, M) over E through p: E → M, and VE := Ker(p*) ⊂ TE is the vertical tangent bundle, a natural vector subbundle of the tangent bundle (TE, πTE, E) of the total space E.

The total space E of any smooth vector bundle carries a natural vector field Vv := vlvv, known as the canonical vector field. More formally, V is a smooth section of (TE, πTE, E), and it can also be defined as the infinitesimal generator of the <a href="Lie_group_action" title="Lie group action">Lie-group action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t,v)\mapsto e^{tv}}">

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<annotation encoding="application/x-tex">{\displaystyle (t,v)\mapsto e^{tv}}</annotation>

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</math><img src="./2270520be288b87abfe11ad21428a258ca82b48d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.131ex; height:3.009ex;" alt="{\displaystyle (t,v)\mapsto e^{tv}}" loading="lazy"> given by the fibrewise scalar multiplication. The canonical vector field V characterizes completely the smooth vector bundle structure in the following manner. As a preparation, note that when X is a smooth vector field on a smooth manifold M and x ∈ M such that Xx = 0, the linear mapping

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<annotation encoding="application/x-tex">{\displaystyle C_{x}(X):T_{x}M\to T_{x}M;\quad C_{x}(X)Y=(\nabla _{Y}X)_{x}}</annotation>

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does not depend on the choice of the linear <a href="Covariant_derivative" title="Covariant derivative">covariant derivative</a> ∇ on M. The canonical vector field V on E satisfies the axioms

<ol><li>The flow (t, v) → ΦtV(v) of V is globally defined.</li>

<li>For each v ∈ V there is a unique limt→∞ ΦtV(v) ∈ V.</li>

<li>Cv(V)∘Cv(V) = Cv(V) whenever Vv = 0.</li>

<li>The <a href="Zero_set" class="mw-redirect" title="Zero set">zero set</a> of V is a smooth <a href="Submanifold" title="Submanifold">submanifold</a> of E whose <a href="Codimension" title="Codimension">codimension</a> is equal to the rank of Cv(V).</li></ol>

Conversely, if E is any smooth manifold and V is a smooth vector field on E satisfying 1–4, then there is a unique vector bundle structure on E whose canonical vector field is V.

For any smooth vector bundle (E, p, M) the total space TE of its tangent bundle (TE, πTE, E) has a natural <a href="Secondary_vector_bundle_structure" title="Secondary vector bundle structure">secondary vector bundle structure</a> (TE, p*, TM), where p* is the <a href="Pushforward_(differential)" title="Pushforward (differential)">push-forward</a> of the canonical projection p: E → M. The vector bundle operations in this secondary vector bundle structure are the push-forwards +*: T(E × E) → TE and λ*: TE → TE of the original addition +: E × E → E and scalar multiplication λ: E → E.

<div class="mw-heading mw-heading2"><h2 id="K-theory">K-theory</h2></div>

The K-theory group, K(X), of a compact <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a> topological space is defined as the <a href="Abelian_group" title="Abelian group">abelian group</a> generated by <a href="Isomorphism_class" class="mw-redirect" title="Isomorphism class">isomorphism classes</a> [E] of <a href="Complex_vector_bundle" title="Complex vector bundle">complex vector bundles</a> under the group operation of <a href="Whitney_sum" class="mw-redirect" title="Whitney sum">Whitney sum</a>, modulo the <a href="Relation_(mathematics)" title="Relation (mathematics)">relation</a> that, whenever we have an <a href="Exact_sequence" title="Exact sequence">exact sequence</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\to A\to B\to C\to 0,}</annotation>

</semantics>

</math>

then

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [B]=[A]+[C]}</annotation>

</semantics>

</math> in <a href="Topological_K-theory" title="Topological K-theory">topological K-theory</a>. <a href="KO-theory" class="mw-redirect" title="KO-theory">KO-theory</a> is a version of this construction which considers real vector bundles. K-theory with <a href="Compact_support" class="mw-redirect" title="Compact support">compact supports</a> can also be defined, as well as higher K-theory groups.

The famous <a href="Bott_periodicity" class="mw-redirect" title="Bott periodicity">periodicity theorem</a> of <a href="Raoul_Bott" title="Raoul Bott">Raoul Bott</a> asserts that the K-theory of any space X is isomorphic to that of the S2X, the double suspension of X.

In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, one considers the K-theory groups consisting of <a href="Coherent_sheaf" title="Coherent sheaf">coherent sheaves</a> on a <a href="Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> X, as well as the K-theory groups of vector bundles on the scheme with the above <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a>. The two constructs are naturally isomorphic provided that the underlying scheme is <a href="Smooth_morphism" title="Smooth morphism">smooth</a>.

<div class="mw-heading mw-heading3"><h3 id="General_notions">General notions</h3></div>

<ul><li><a href="Grassmannian" title="Grassmannian">Grassmannian</a>: <a href="Classifying_space" title="Classifying space">classifying spaces</a> for vector bundle, among which <a href="Projective_space" title="Projective space">projective spaces</a> for <a href="Line_bundle" title="Line bundle">line bundles</a></li>

<li><a href="Characteristic_class" title="Characteristic class">Characteristic class</a></li>

<li><a href="Splitting_principle" title="Splitting principle">Splitting principle</a></li>

<li><a href="Stable_bundle" class="mw-redirect" title="Stable bundle">Stable bundle</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Topology_and_differential_geometry">Topology and differential geometry</h3></div>

<ul><li><a href="Connection_(vector_bundle)" title="Connection (vector bundle)">Connection</a>: the notion needed to differentiate sections of vector bundles.</li>

<li><a href="Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a>: the general study of connections on vector bundles and principal bundles and their relations to physics.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Algebraic_and_analytic_geometry">Algebraic and analytic geometry</h3></div>

<ul><li><a href="Algebraic_vector_bundle" class="mw-redirect" title="Algebraic vector bundle">Algebraic vector bundle</a></li>

<li><a href="Picard_group" title="Picard group">Picard group</a></li>

<li><a href="Holomorphic_vector_bundle" title="Holomorphic vector bundle">Holomorphic vector bundle</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist">

<li id="cite_note-FOOTNOTEHatcher2003Example_3.6-1"><a href="#cite_ref-FOOTNOTEHatcher2003Example_3.6_1-0">^</a> <a href="#CITEREFHatcher2003">Hatcher 2003</a>, Example 3.6.

</li>

<li id="cite_note-FOOTNOTELang1995-2"><a href="#cite_ref-FOOTNOTELang1995_2-0">^</a> <a href="#CITEREFLang1995">Lang 1995</a>.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2></div>

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFAbrahamMarsden1978" class="citation cs2"><a href="Ralph_Abraham_(mathematician)" title="Ralph Abraham (mathematician)">Abraham, Ralph H.</a>; <a href="Jerrold_E._Marsden" title="Jerrold E. Marsden">Marsden, Jerrold E.</a> (1978), Foundations of mechanics, London: Benjamin-Cummings, see section 1.5, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978fome.book.....A">1978fome.book.....A</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8053-0102-1</bdi></cite>.</li>

<li><cite id="CITEREFHatcher2003" class="citation cs2"><a href="Allen_Hatcher" title="Allen Hatcher">Hatcher, Allen</a> (2003), <a rel="nofollow" class="external text" href="https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html">Vector Bundles & K-Theory</a> (2.0 ed.)</cite>.</li>

<li><cite id="CITEREFJost2002" class="citation cs2">Jost, Jürgen (2002), Riemannian Geometry and Geometric Analysis (3rd ed.), Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-42627-1</bdi></cite>, see section 1.5.</li>

<li><cite id="CITEREFLang1995" class="citation cs2"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (1995), Differential and Riemannian manifolds, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-94338-1</bdi></cite>.</li>

<li><cite id="CITEREFLee2009" class="citation cs2">Lee, Jeffrey M. (2009), <a rel="nofollow" class="external text" href="http://www.ams.org/bookstore-getitem/item=gsm-107">Manifolds and Differential Geometry</a>, <a href="Graduate_Studies_in_Mathematics" title="Graduate Studies in Mathematics">Graduate Studies in Mathematics</a>, vol. 107, Providence: American Mathematical Society, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-4815-9</bdi></cite>.</li>

<li><cite id="CITEREFLee2003" class="citation cs2">Lee, John M. (2003), <a rel="nofollow" class="external text" href="http://www.math.washington.edu/~lee/Books/smooth.html">Introduction to Smooth Manifolds</a>, New York: Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-95448-1</bdi></cite> see Ch.5</li>

<li><cite id="CITEREFRubei2014" class="citation cs2">Rubei, Elena (2014), Algebraic Geometry, a concise dictionary, Berlin/Boston: Walter De Gruyter, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-11-031622-3</bdi></cite>.</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Vector_bundle">"Vector bundle"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/7836">Why is it useful to study vector bundles ?</a> on <a href="MathOverflow" title="MathOverflow">MathOverflow</a></li>

<li><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/16240">Why is it useful to classify the vector bundles of a space ?</a></li></ul>

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</style><div id="Manifolds_(Glossary,_List,_Category)274" style="font-size:114%;margin:0 4em"><a href="Manifold" title="Manifold">Manifolds</a> (<a href="Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>, <a href="List_of_manifolds" title="List of manifolds">List</a>, Category)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Topological_manifold" title="Topological manifold">Topological manifold</a>

<ul><li><a href="Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li>

<li><a href="Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a>

<ul><li><a href="Differential_structure" title="Differential structure">Differential structure</a></li>

<li><a href="Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li>

<li><a href="Submanifold" title="Submanifold">Submanifold</a></li>

<li><a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li>

<li><a href="Smoothness" title="Smoothness">Smooth map</a></li>

<li><a href="Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li>

<li><a href="Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li>

<li><a href="Tangent_space" title="Tangent space">Tangent space</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a></li>

<li><a href="Vector_field" title="Vector field">Vector field</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results (list)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li>

<li><a href="Darboux's_theorem" title="Darboux's theorem">Darboux's</a></li>

<li><a href="Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li>

<li><a href="Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li>

<li><a href="Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li>

<li><a href="Noether's_theorem" title="Noether's theorem">Noether's</a></li>

<li><a href="Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Smoothness" title="Smoothness">Maps</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Differentiable_curve" title="Differentiable curve">Curve</a></li>

<li><a href="Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a>

<ul><li><a href="Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li>

<li><a href="Geodesic" title="Geodesic">Geodesic</a></li>

<li><a href="Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a>

<ul><li><a href="Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li>

<li><a href="Foliation" title="Foliation">Foliation</a></li>

<li><a href="Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li>

<li><a href="Integral_curve" title="Integral curve">Integral curve</a></li>

<li><a href="Lie_derivative" title="Lie derivative">Lie derivative</a></li>

<li><a href="Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li>

<li><a href="Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of manifolds</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Closed_manifold" title="Closed manifold">Closed</a></li>

<li><a href="Collapsing_manifold" title="Collapsing manifold">Collapsing</a></li>

<li><a href="Complete_manifold" title="Complete manifold">Complete</a></li>

<li>(<a href="Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="Complex_manifold" title="Complex manifold">Complex</a></li>

<li>(<a href="Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li>

<li><a href="Fibered_manifold" title="Fibered manifold">Fibered</a></li>

<li><a href="Finsler_manifold" title="Finsler manifold">Finsler</a></li>

<li>(<a href="Almost_flat_manifold" title="Almost flat manifold">Almost</a>) <a href="Flat_manifold" title="Flat manifold">Flat</a></li>

<li><a href="G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li>

<li><a href="Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li>

<li><a href="Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li>

<li><a href="Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li>

<li><a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li>

<li><a href="Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li>

<li><a href="Lie_group" title="Lie group">Lie group</a>

<ul><li><a href="Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li>

<li><a href="Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li>

<li><a href="Nilmanifold" title="Nilmanifold">Nilmanifold</a></li>

<li><a href="Orientability" title="Orientability">Oriented</a></li>

<li><a href="Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li>

<li><a href="Poisson_manifold" title="Poisson manifold">Poisson</a></li>

<li><a href="Prime_manifold" title="Prime manifold">Prime</a></li>

<li><a href="Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li>

<li><a href="Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li>

<li>(<a href="Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-</a>, <a href="Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub-</a>) <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li>

<li><a href="Rizza_manifold" title="Rizza manifold">Rizza</a></li>

<li><a href="Stein_manifold" title="Stein manifold">Stein</a></li>

<li>(<a href="Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li>

<li><a href="Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li>

<li><a href="Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li>

<li><a href="Tangent_space" title="Tangent space">Tangent space</a>

<ul><li><a href="Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li>

<li><a href="Torsion_tensor" title="Torsion tensor">Torsion</a></li>

<li><a href="Vector_field" title="Vector field">Vector field</a></li>

<li><a href="Vector_flow" title="Vector flow">Vector flow</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li>

<li><a href="Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li>

<li><a href="Cotangent_space" title="Cotangent space">Cotangent space</a>

<ul><li><a href="Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li>

<li><a href="De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a>

<ul><li><a href="Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Interior_product" title="Interior product">Interior product</a></li>

<li><a href="Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li>

<li><a href="Ricci_curvature" title="Ricci curvature">Ricci curvature</a>

<li><a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li>

<li><a href="Tensor_field" title="Tensor field">Tensor field</a>

<ul><li><a href="Tensor_density" title="Tensor density">density</a></li></ul></li>

<li><a href="Volume_form" title="Volume form">Volume form</a></li>

<li><a href="Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li>

<li><a href="Affine_bundle" title="Affine bundle">Affine</a></li>

<li><a href="Associated_bundle" title="Associated bundle">Associated</a></li>

<li><a href="Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li>

<li><a href="Fiber_bundle" title="Fiber bundle">Fiber</a></li>

<li>(<a href="Cofibration" title="Cofibration">Co-</a>) <a href="Fibration" title="Fibration">Fibration</a></li>

<li><a href="Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li>

<li>(<a href="Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="Normal_bundle" title="Normal bundle">Normal</a></li>

<li><a href="Principal_bundle" title="Principal bundle">Principal</a></li>

<li><a href="Spinor_bundle" title="Spinor bundle">Spinor</a></li>

<li><a href="Subbundle" title="Subbundle">Subbundle</a></li>

<li><a href="Tangent_bundle" title="Tangent bundle">Tangent</a></li>

<li><a href="Tensor_bundle" title="Tensor bundle">Tensor</a></li>

</ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Connection_(mathematics)" title="Connection (mathematics)">Connections</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_connection" title="Affine connection">Affine</a></li>

<li><a href="Cartan_connection" title="Cartan connection">Cartan</a></li>

<li><a href="Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li>

<li><a href="Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li>

<li><a href="Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li>

<li><a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li>

<li><a href="Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li>

<li><a href="Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li>

<li><a href="Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li>

<li><a href="Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li>

<li><a href="History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li>

<li><a href="Morse_theory" title="Morse theory">Morse theory</a></li>

<li><a href="Moving_frame" title="Moving frame">Moving frame</a></li>

<li><a href="Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_manifold" title="Banach manifold">Banach manifold</a></li>

<li><a href="Diffeology" title="Diffeology">Diffeology</a></li>

<li><a href="Diffiety" title="Diffiety">Diffiety</a></li>

<li><a href="Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li>

<li><a href="K-theory" title="K-theory">K-theory</a></li>

<li><a href="Orbifold" title="Orbifold">Orbifold</a></li>

<li><a href="Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a>

<ul><li><a href="Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li>

<li><a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li>

<li><a href="Stratifold" title="Stratifold">Stratifold</a></li>

<li><a href="Supermanifold" title="Supermanifold">Supermanifold</a></li>

<li><a href="Stratified_space" title="Stratified space">Stratified space</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-23" href="https://en.wikipedia.org/wiki/?title=Vector_bundle&oldid=1302140056">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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