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Natural bundle

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Differential_geometry" title="Differential geometry">differential geometry</a>, a field in <a href="Mathematics" title="Mathematics">mathematics</a>, a natural bundle is any <a href="Fiber_bundle" title="Fiber bundle">fiber bundle</a> associated to the s-frame bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{s}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle F^{s}(M)}</annotation>

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</math><img src="./e42cbd7da8f8fa781fbc78a0be2e2cd0873a8609.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.07ex; height:2.843ex;" alt="{\displaystyle F^{s}(M)}" loading="lazy"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\geq 1}">

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<annotation encoding="application/x-tex">{\displaystyle s\geq 1}</annotation>

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</math><img src="./384d582d5fd523b9cbb7f85c5fb7d096cf478e9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.351ex; height:2.343ex;" alt="{\displaystyle s\geq 1}" loading="lazy">. It turns out that its transition functions depend functionally on local changes of coordinates in the base <a href="Manifold" title="Manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> together with their partial derivatives up to order at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy">.<a href="#cite_note-1">[1]</a>

The concept of a natural bundle was introduced by <a href="Albert_Nijenhuis" title="Albert Nijenhuis">Albert Nijenhuis</a> as a modern reformulation of the classical concept of an arbitrary bundle of geometric objects.<a href="#cite_note-2">[2]</a>

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

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Mf}">

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

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</math><img src="./9c953a4820cd670853f2057aab0b24393f7ad5eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.721ex; height:2.509ex;" alt="{\displaystyle Mf}" loading="lazy"> denote the <a href="Category_of_manifolds" title="Category of manifolds">category of smooth manifolds</a> and <a href="Smooth_map" class="mw-redirect" title="Smooth map">smooth maps</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Mf_{n}}">

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

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</math><img src="./f5ec03054c4f019214c02a82fad73b6371342a00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.8ex; height:2.509ex;" alt="{\displaystyle Mf_{n}}" loading="lazy"> the <a href="Category_(mathematics)" title="Category (mathematics)">category</a> of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensional manifolds and <a href="Local_diffeomorphism" title="Local diffeomorphism">local diffeomorphisms</a>. Consider also the category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {FM}}}">

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

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</math><img src="./99b1227f3c21b13496af3a03b8be41c4917da042.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.717ex; height:2.176ex;" alt="{\displaystyle {\mathcal {FM}}}" loading="lazy"> of <a href="Fibered_manifold" title="Fibered manifold">fibred manifolds</a> and bundle morphisms, and the functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B:{\mathcal {FM}}\to {\mathcal {M}}f}">

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<annotation encoding="application/x-tex">{\displaystyle B:{\mathcal {FM}}\to {\mathcal {M}}f}</annotation>

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</math><img src="./e72a1d9e2616f377fea2f6020d729a84fb1357d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.102ex; height:2.509ex;" alt="{\displaystyle B:{\mathcal {FM}}\to {\mathcal {M}}f}" loading="lazy"> associating to any fibred manifold its base manifold.

A natural bundle (or bundle functor) is a <a href="Functor" title="Functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:{\mathcal {M}}f_{n}\to {\mathcal {FM}}}">

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<annotation encoding="application/x-tex">{\displaystyle F:{\mathcal {M}}f_{n}\to {\mathcal {FM}}}</annotation>

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</math><img src="./a916d0d159b6c1c58290ec190645b60b4d554bd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.157ex; height:2.509ex;" alt="{\displaystyle F:{\mathcal {M}}f_{n}\to {\mathcal {FM}}}" loading="lazy"> satisfying the following three properties:

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

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</math><img src="./2685ba3a81eb4e80afcd05eec31c9b1e71bbddf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.737ex; height:2.176ex;" alt="{\displaystyle B\circ F=\mathrm {id} }" loading="lazy">, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(M)}">

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

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</math><img src="./5a3c995b733d7afe3ed460a7dab375905c022674.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.016ex; height:2.843ex;" alt="{\displaystyle B(M)}" loading="lazy"> is a fibred manifold over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, with projection denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{M}:B(M)\to M}">

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<annotation encoding="application/x-tex">{\displaystyle p_{M}:B(M)\to M}</annotation>

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</math><img src="./a00972fe3ec584be0847ac688dd851a087a928db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:17.227ex; height:2.843ex;" alt="{\displaystyle p_{M}:B(M)\to M}" loading="lazy">;</li>

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

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</math><img src="./a57ecb017c8458e5f519dbca5ddb7fffde7f1a16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.323ex; height:2.343ex;" alt="{\displaystyle U\subseteq M}" loading="lazy"> is an open <a href="Submanifold" title="Submanifold">submanifold</a>, with inclusion map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:U\hookrightarrow M}">

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

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</math><img src="./24d56df28d7346ef80d04c1d1d57f5bca5598649.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.871ex; height:2.176ex;" alt="{\displaystyle i:U\hookrightarrow M}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(U)}">

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

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</math><img src="./462254eb7da92fe91d24532826f4e036cf1eeaa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.333ex; height:2.843ex;" alt="{\displaystyle F(U)}" loading="lazy"> coincides with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{M}^{-1}(U)\subseteq F(M)}">

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<annotation encoding="application/x-tex">{\displaystyle p_{M}^{-1}(U)\subseteq F(M)}</annotation>

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</math><img src="./d4993aa662c10f13611a35f508e8dfc66298db98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:16.274ex; height:3.343ex;" alt="{\displaystyle p_{M}^{-1}(U)\subseteq F(M)}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(i):F(U)\to F(M)}">

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<annotation encoding="application/x-tex">{\displaystyle F(i):F(U)\to F(M)}</annotation>

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</math><img src="./999af6528567852cfeed106a38f378138a708da1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.229ex; height:2.843ex;" alt="{\displaystyle F(i):F(U)\to F(M)}" loading="lazy"> is the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{-1}(U)\hookrightarrow F(M)}">

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<annotation encoding="application/x-tex">{\displaystyle p^{-1}(U)\hookrightarrow F(M)}</annotation>

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</math><img src="./511c72bb3654739b0ef9f36e846bc0171fb019c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:17.083ex; height:3.176ex;" alt="{\displaystyle p^{-1}(U)\hookrightarrow F(M)}" loading="lazy">;</li>

<li>for any smooth map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:P\times M\to N}">

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<annotation encoding="application/x-tex">{\displaystyle f:P\times M\to N}</annotation>

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</math><img src="./709d9a6e11c8e04f570f3b49e28078d2a082dff3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.921ex; height:2.509ex;" alt="{\displaystyle f:P\times M\to N}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(p,\cdot ):M\to N}">

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<annotation encoding="application/x-tex">{\displaystyle f(p,\cdot ):M\to N}</annotation>

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</math><img src="./4995d5f29bcb873285457f65a8af1b309f6b9d18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.995ex; height:2.843ex;" alt="{\displaystyle f(p,\cdot ):M\to N}" loading="lazy"> is a local diffeomorphism for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in P}">

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

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</math><img src="./f4dc1da33d78a2bdae06c9edff726feb126dae34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.845ex; height:2.509ex;" alt="{\displaystyle p\in P}" loading="lazy">, then the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\times F(M)\to F(N),(p,x)\mapsto F(f(p,\cdot ))(x)}">

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<annotation encoding="application/x-tex">{\displaystyle P\times F(M)\to F(N),(p,x)\mapsto F(f(p,\cdot ))(x)}</annotation>

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</math><img src="./8766265cc72400b6e9d55b602758404a4e1ed3a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.423ex; height:2.843ex;" alt="{\displaystyle P\times F(M)\to F(N),(p,x)\mapsto F(f(p,\cdot ))(x)}" loading="lazy"> is smooth.</li></ol>

As a consequence of the first condition, one has a <a href="Natural_transformation" title="Natural transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:F\to B}">

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

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</math><img src="./f8c91871d751d9d9b1714e0e9b48006592128fa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle p:F\to B}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Finite_order_natural_bundles">Finite order natural bundles</h2></div>

A natural bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:Mf_{n}\to Mf}">

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<annotation encoding="application/x-tex">{\displaystyle F:Mf_{n}\to Mf}</annotation>

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</math><img src="./b0ca5a2b0785590be404dfea74449573a74f1a54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.813ex; height:2.509ex;" alt="{\displaystyle F:Mf_{n}\to Mf}" loading="lazy"> is called of finite order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> if, for every local diffeomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:M\to N}">

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<annotation encoding="application/x-tex">{\displaystyle f:M\to N}</annotation>

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</math><img src="./1dbd50e2de9728ee14a7c232441137f588b109f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.336ex; height:2.509ex;" alt="{\displaystyle f:M\to N}" loading="lazy"> and every point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M}">

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

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</math><img src="./9df57d73e9532bb93a1439890bcddbc2806f5859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}" loading="lazy">, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(f)_{x}:F(M)_{x}\to F(N)_{f(x)}}">

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<annotation encoding="application/x-tex">{\displaystyle F(f)_{x}:F(M)_{x}\to F(N)_{f(x)}}</annotation>

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</math><img src="./18580e1c5dbc72f2e55f8d42a63abbbf7f6e35c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.687ex; height:3.176ex;" alt="{\displaystyle F(f)_{x}:F(M)_{x}\to F(N)_{f(x)}}" loading="lazy"> depends only on the jet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{x}^{r}f}">

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<annotation encoding="application/x-tex">{\displaystyle j_{x}^{r}f}</annotation>

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</math><img src="./ebafea4346cca97aa1eb0a2fca7c56ea47d33a34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:3.436ex; height:2.509ex;" alt="{\displaystyle j_{x}^{r}f}" loading="lazy">. Equivalently, for every local diffeomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g:M\to N}">

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<annotation encoding="application/x-tex">{\displaystyle f,g:M\to N}</annotation>

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</math><img src="./5e5ee910aa0d10f90847bdadbea5ddc8680a30c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.486ex; height:2.509ex;" alt="{\displaystyle f,g:M\to N}" loading="lazy"> and every point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M}">

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

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</math><img src="./9df57d73e9532bb93a1439890bcddbc2806f5859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}" loading="lazy">, one has<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{x}^{r}f=j_{x}^{r}g\Rightarrow F(f)|_{F(M)_{x}}=F(g)|_{F(M)_{x}}.}">

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<annotation encoding="application/x-tex">{\displaystyle j_{x}^{r}f=j_{x}^{r}g\Rightarrow F(f)|_{F(M)_{x}}=F(g)|_{F(M)_{x}}.}</annotation>

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</math>Natural bundles of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> coincide with the associated fibre bundles to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">-th order <a href="Frame_bundle" title="Frame bundle">frame bundles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{s}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle F^{s}(M)}</annotation>

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</math><img src="./e42cbd7da8f8fa781fbc78a0be2e2cd0873a8609.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.07ex; height:2.843ex;" alt="{\displaystyle F^{s}(M)}" loading="lazy">.

A classical result by <a href="David_B._A._Epstein" title="David B. A. Epstein">Epstein</a> and <a href="William_Thurston" title="William Thurston">Thurston</a> shows that all natural bundles have finite order.<a href="#cite_note-3">[3]</a>

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

An example of natural bundle (of first order) is the <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TM}">

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

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</math><img src="./ea000afb5769206ddd5fd43f458430d04422ddeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.078ex; height:2.176ex;" alt="{\displaystyle TM}" loading="lazy"> of a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">.

Other examples include the cotangent bundles, the bundles of metrics of signature <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,s)}">

</mstyle>

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

</semantics>

</math><img src="./2f3e3778daed31bfa6149169529acb5d9fcc8b49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.982ex; height:2.843ex;" alt="{\displaystyle (r,s)}" loading="lazy"> and the bundle of linear connections.<a href="#cite_note-4">[4]</a>

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

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</style><cite id="CITEREFPalaisTerng1977" class="citation cs2"><a href="Richard_Palais" title="Richard Palais">Palais, Richard</a>; <a href="Chuu-Lian_Terng" title="Chuu-Lian Terng">Terng, Chuu-Lian</a> (1977), "Natural bundles have finite order", Topology, 16 (3): 271–277, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0040-9383%2877%2990008-8">10.1016/0040-9383(77)90008-8</a>, <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F102222">10338.dmlcz/102222</a></cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFA._Nijenhuis1972" class="citation cs2"><a href="Albert_Nijenhuis" title="Albert Nijenhuis">A. Nijenhuis</a> (1972), Natural bundles and their general properties, Tokyo: Diff. Geom. in Honour of K. Yano, pp. 317–334</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFEpsteinThurston1979" class="citation journal cs1 cs1-prop-long-vol"><a href="David_B._A._Epstein" title="David B. A. Epstein">Epstein, D. B. A.</a>; <a href="William_Thurston" title="William Thurston">Thurston, W. P.</a> (1979). <a rel="nofollow" class="external text" href="http://doi.wiley.com/10.1112/plms/s3-38.2.219">"Transformation Groups and Natural Bundles"</a>. <a href="Proceedings_of_the_London_Mathematical_Society" class="mw-redirect" title="Proceedings of the London Mathematical Society">Proceedings of the London Mathematical Society</a>. s3-38 (2): 219–236. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs3-38.2.219">10.1112/plms/s3-38.2.219</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFFatibeneFrancaviglia2003" class="citation book cs1">Fatibene, Lorenzo; <a href="Mauro_Francaviglia" title="Mauro Francaviglia">Francaviglia, Mauro</a> (2003). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-94-017-2384-8">Natural and Gauge Natural Formalism for Classical Field Theorie</a>. Springer. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-017-2384-8">10.1007/978-94-017-2384-8</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4020-1703-2</bdi>.</cite>

</li>

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

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

<ul><li><cite id="CITEREFKolářMichorSlovák1993" class="citation cs2">Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170330154524/http://www.emis.de/monographs/KSM/kmsbookh.pdf">Natural operators in differential geometry</a> (PDF), Springer-Verlag, archived from <a rel="nofollow" class="external text" href="http://www.emis.de/monographs/KSM/kmsbookh.pdf">the original</a> (PDF) on 2017-03-30, retrieved 2017-08-15</cite></li>

<li><cite id="CITEREFKrupkaJanyška1990" class="citation cs2">Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>80-210-0165-8</bdi></cite></li>

<li><cite id="CITEREFSaunders1989" class="citation cs2">Saunders, D.J. (1989), <a rel="nofollow" class="external text" href="https://archive.org/details/geometryofjetbun0000saun">The geometry of jet bundles</a>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-36948-7</bdi></cite></li></ul></div><div><div class="zim-footer">

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