About: Unit tangent bundle

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In Riemannian geometry, the unit tangent bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M) or simply UTM, is the unit sphere bundle for the tangent bundle T(M). It is a fiber bundle over M whose fiber at each point is the unit sphere in the tangent bundle: where Tx(M) denotes the tangent space 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 projection

Attributes	Values
rdf:type	yago:AnimalTissue105267548 yago:BodyPart105220461 yago:FiberBundle105475681 yago:NervousTissue105296775 yago:Part109385911 yago:PhysicalEntity100001930 yago:Thing100002452 yago:Tissue105267345 yago:WikicatFiberBundles
rdfs:label	Einheits-Tangentialbündel (de) Fibré tangent unitaire (fr) Unit tangent bundle (en)
rdfs:comment	In der Mathematik bezeichnet das Einheits-Tangentialbündel den Raum aller Tangentialvektoren der Länge 1 zu einer gegebenen Mannigfaltigkeit, zum Beispiel zu einer Fläche im . Der Begriff spielt eine wichtige Rolle in der Differentialgeometrie und der Theorie der dynamischen Systeme. (de) En géométrie différentielle, le fibré tangent unitaire d'une variété riemannienne est l'ensemble des vecteurs de norme 1 de son fibré tangent, muni de la topologie induite. Le fibré tangent unitaire est important car c'est sur lui qu'agit le flot géodésique. * Portail de la géométrie (fr) In Riemannian geometry, the unit tangent bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M) or simply UTM, is the unit sphere bundle for the tangent bundle T(M). It is a fiber bundle over M whose fiber at each point is the unit sphere in the tangent bundle: where Tx(M) denotes the tangent space 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 projection (en)
dcterms:subject	Differential topology Riemannian geometry Ergodic theory Fiber bundles
Wikipage page ID	7932644 (xsd:integer)
Wikipage revision ID	980098771 (xsd:integer)
Link from a Wikipage to another Wikipage	Pushforward (differential) Volume element Volume form Measure (mathematics) Tangent bundle Geodesic Levi-Civita connection Linear isomorphism Fréchet manifold Tangent space Unit sphere Differential topology Euclidean space Fiber bundle Finsler manifold Banach manifold Hilbert manifold Projection (mathematics) Radon measure Riemannian geometry Riemannian manifold Riemannian geometry Ergodic theory Fiber bundles Jürgen Jost Hessian matrix Tautological one-form Borel measure Jerrold Marsden Ralph Abraham (mathematician) Hypersphere Sasakian manifold Contact structure
sameAs	Unit tangent bundle Unit tangent bundle Unit tangent bundle Unit tangent bundle Unit tangent bundle Unit tangent bundle
dbp:wikiPageUsesTemplate	dbt:ISBN dbt:No_inline_citations
has abstract	In der Mathematik bezeichnet das Einheits-Tangentialbündel den Raum aller Tangentialvektoren der Länge 1 zu einer gegebenen Mannigfaltigkeit, zum Beispiel zu einer Fläche im . Der Begriff spielt eine wichtige Rolle in der Differentialgeometrie und der Theorie der dynamischen Systeme. (de) En géométrie différentielle, le fibré tangent unitaire d'une variété riemannienne est l'ensemble des vecteurs de norme 1 de son fibré tangent, muni de la topologie induite. Le fibré tangent unitaire est important car c'est sur lui qu'agit le flot géodésique. * Portail de la géométrie (fr) In Riemannian geometry, the unit tangent bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M) or simply UTM, is the unit sphere bundle for the tangent bundle T(M). It is a fiber bundle over M whose fiber at each point is the unit sphere in the tangent bundle: where Tx(M) denotes the tangent space 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 projection 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)-sphere Sn−1, where n is the dimension of M. The unit tangent bundle is therefore a sphere bundle over M with fiber Sn−1. The definition of unit sphere bundle can easily accommodate Finsler manifolds 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: If M is an infinite-dimensional manifold (for example, a Banach, Fréchet or Hilbert manifold), 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. (en)
prov:wasDerivedFrom	wikipedia-en:Unit_tangent_bundle?oldid=980098771&ns=0
page length (characters) of wiki page	5584 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Unit_tangent_bundle
is Link from a Wikipage to another Wikipage of	Tangent bundle Quantum ergodicity SL2(R) Geodesic Fiber bundle Poincaré half-plane model Stiefel manifold Unit sphere bundle
is Wikipage redirect of	Unit sphere bundle
is foaf:primaryTopic of	wikipedia-en:Unit_tangent_bundle

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