SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Unit tangent bundle</title>

</head>

Unit tangent bundle

</h1>

</header>

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

.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}

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

</style>

In <a href="Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, the unit tangent bundle of a <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> (M, g), denoted by T1M, UT(M), UTM, or SM is the unit sphere bundle for the <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> T(M). It is a <a href="Fiber_bundle" title="Fiber bundle">fiber bundle</a> over M whose fiber at each point is the <a href="Unit_sphere" title="Unit sphere">unit sphere</a> in the tangent space:

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {UT} (M):=\coprod _{x\in M}\left\{v\in \mathrm {T} _{x}(M)\left|g_{x}(v,v)=1\right.\right\},}</annotation>

</semantics>

</math><img src="./92d097f3e97e3e92a35359beba254df638406c23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.672ex; height:5.676ex;" alt="{\displaystyle \mathrm {UT} (M):=\coprod _{x\in M}\left\{v\in \mathrm {T} _{x}(M)\left|g_{x}(v,v)=1\right.\right\},}" loading="lazy"></dd></dl>

where Tx(M) denotes the <a href="Tangent_space" title="Tangent space">tangent space</a> to M at x. Thus, elements of UT(M) are pairs (x, v), where x is some point of the manifold and v is some tangent direction (of unit length) to the manifold at x. The unit tangent bundle is equipped with a natural <a href="Projection_(mathematics)" title="Projection (mathematics)">projection</a>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi :\mathrm {UT} (M)\to M,}</annotation>

</semantics>

</math><img src="./522466e8163b912a9624a75e64f2fdc667dc6086.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.645ex; height:2.843ex;" alt="{\displaystyle \pi :\mathrm {UT} (M)\to M,}" loading="lazy"></dd>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./312808fc8e12775f372abe59c8369db3fe1beb0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.16ex; height:2.843ex;" alt="{\displaystyle \pi :(x,v)\mapsto x,}" loading="lazy"></dd></dl>

which takes each point of the bundle to its base point. The fiber π−1(x) over each point x ∈ M is an (n−1)-<a href="Hypersphere" class="mw-redirect" title="Hypersphere">sphere</a> Sn−1, where n is the dimension of M. The unit tangent bundle is therefore a <a href="Fiber_bundle#Sphere_bundles" title="Fiber bundle">sphere bundle</a> over M with fiber Sn−1.

The definition of unit sphere bundle can easily accommodate <a href="Finsler_manifold" title="Finsler manifold">Finsler manifolds</a> as well. Specifically, if M is a manifold equipped with a Finsler metric F : TM → R, then the unit sphere bundle is the subbundle of the tangent bundle whose fiber at x is the indicatrix of F:

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {UT} _{x}(M)=\left\{v\in \mathrm {T} _{x}(M)\left|F(v)=1\right.\right\}.}</annotation>

</semantics>

</math><img src="./1435f8fbc6bfdacc8afc2720bc248b3324d969ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.345ex; height:2.843ex;" alt="{\displaystyle \mathrm {UT} _{x}(M)=\left\{v\in \mathrm {T} _{x}(M)\left|F(v)=1\right.\right\}.}" loading="lazy"></dd></dl>

If M is an infinite-dimensional manifold (for example, a <a href="Banach_manifold" title="Banach manifold">Banach</a>, <a href="Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet</a> or <a href="Hilbert_manifold" title="Hilbert manifold">Hilbert manifold</a>), then UT(M) can still be thought of as the unit sphere bundle for the tangent bundle T(M), but the fiber π−1(x) over x is then the infinite-dimensional unit sphere in the tangent space.

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

The unit tangent bundle carries a variety of differential geometric structures. The metric on M induces a <a href="Contact_structure" class="mw-redirect" title="Contact structure">contact structure</a> on UTM. This is given in terms of a <a href="Tautological_one-form" title="Tautological one-form">tautological one-form</a>, defined at a point u of UTM (a unit tangent vector of M) by

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{u}(v)=g(u,\pi _{*}v)\,}</annotation>

</semantics>

</math><img src="./daa50a0df5744047b7d4f8abcbae216119266647.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.481ex; height:2.843ex;" alt="{\displaystyle \theta _{u}(v)=g(u,\pi _{*}v)\,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{*}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8b557457bc4b3bfcec162d7ce45c4edacec83b9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:2.379ex; height:2.009ex;" alt="{\displaystyle \pi _{*}}" loading="lazy"> is the <a href="Pushforward_(differential)" title="Pushforward (differential)">pushforward</a> along π of the vector v ∈ TuUTM.

Geometrically, this contact structure can be regarded as the distribution of (2n−2)-planes which, at the unit vector u, is the pullback of the orthogonal complement of u in the tangent space of M. This is a contact structure, for the fiber of UTM is obviously an integral manifold (the vertical bundle is everywhere in the kernel of θ), and the remaining tangent directions are filled out by moving up the fiber of UTM. Thus the maximal integral manifold of θ is (an open set of) M itself.

On a Finsler manifold, the contact form is defined by the analogous formula

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{u}(v)=g_{u}(u,\pi _{*}v)\,}</annotation>

</semantics>

</math><img src="./ebf68b87ea8374ac748c4d91b1d604ae57a13bd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.647ex; height:2.843ex;" alt="{\displaystyle \theta _{u}(v)=g_{u}(u,\pi _{*}v)\,}" loading="lazy"></dd></dl>

where gu is the fundamental tensor (the <a href="Hessian_matrix" title="Hessian matrix">hessian</a> of the Finsler metric). Geometrically, the associated distribution of hyperplanes at the point u ∈ UTxM is the inverse image under π* of the tangent hyperplane to the unit sphere in TxM at u.

The <a href="Volume_form" title="Volume form">volume form</a> θ∧dθn−1 defines a <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a> on M, known as the kinematic measure, or Liouville measure, that is invariant under the <a href="Geodesic#Geodesic_flow" title="Geodesic">geodesic flow</a> of M. As a <a href="Radon_measure" title="Radon measure">Radon measure</a>, the kinematic measure μ is defined on compactly supported continuous functions ƒ on UTM by

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{UTM}f\,d\mu =\int _{M}dV(p)\int _{UT_{p}M}\left.f\right|_{UT_{p}M}\,d\mu _{p}}</annotation>

</semantics>

</math><img src="./d47a1ad0f9b8f7dbf8c24f1df48a5bbd870bebf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:41.145ex; height:6.176ex;" alt="{\displaystyle \int _{UTM}f\,d\mu =\int _{M}dV(p)\int _{UT_{p}M}\left.f\right|_{UT_{p}M}\,d\mu _{p}}" loading="lazy"></dd></dl>

where dV is the <a href="Volume_element" title="Volume element">volume element</a> on M, and μp is the standard rotationally-invariant <a href="Borel_measure" title="Borel measure">Borel measure</a> on the Euclidean sphere UTpM.

The <a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a> of M gives rise to a splitting of the tangent bundle

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ca525b18758afc7b74fde74e2941af31aaa70af3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.096ex; height:2.843ex;" alt="{\displaystyle T(UTM)=H\oplus V}" loading="lazy"></dd></dl>

into a vertical space V = kerπ* and horizontal space H on which π* is a <a href="Linear_isomorphism" class="mw-redirect" title="Linear isomorphism">linear isomorphism</a> at each point of UTM. This splitting induces a metric on UTM by declaring that this splitting be an orthogonal direct sum, and defining the metric on H by the pullback:

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{H}(v,w)=g(v,w),\quad v,w\in H}</annotation>

</semantics>

</math><img src="./ccbabf19c467c91dcd73ea05ced71e0b5d9a7f73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.371ex; height:2.843ex;" alt="{\displaystyle g_{H}(v,w)=g(v,w),\quad v,w\in H}" loading="lazy"></dd></dl>

and defining the metric on V as the induced metric from the embedding of the fiber UTxM into the <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> TxM. Equipped with this metric and contact form, UTM becomes a <a href="Sasakian_manifold" title="Sasakian manifold">Sasakian manifold</a>.

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

<ul><li>Jeffrey M. Lee: Manifolds and Differential Geometry. Graduate Studies in Mathematics Vol. 107, American Mathematical Society, Providence (2009). <style data-mw-deduplicate="TemplateStyles:r1238218222">

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

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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

</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-4815-9</bdi></li>

<li><a href="J%C3%BCrgen_Jost" title="Jürgen Jost">Jürgen Jost</a>: Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-42627-2</bdi></li>

<li><a href="Ralph_Abraham_(mathematician)" title="Ralph Abraham (mathematician)">Ralph Abraham</a> und <a href="Jerrold_Marsden" class="mw-redirect" title="Jerrold Marsden">Jerrold E. Marsden</a>: Foundations of Mechanics, (1978) Benjamin-Cummings, London. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8053-0102-X</bdi></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-10-10" href="https://en.wikipedia.org/wiki/?title=Unit_tangent_bundle&oldid=1250519633">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.

</div>

</div></div>

</div>

</main>

</div>

</body></html>