Unit tangential bundle

In mathematics , the unit tangent bundle denotes the space of all tangent vectors of length 1 to a given manifold, for example to a surface im . The term plays an important role in differential geometry and the theory of dynamic systems . ${\displaystyle \mathbb {R} ^{3}}$

definition

Let it be a Riemannian manifold and its tangential bundle . The unit tangent bundle is ${\displaystyle (M,g)}$ ${\displaystyle TM}$

${\displaystyle T^{1}M:=\coprod _{x\in M}\left\{v\in T_{x}(M)\left|g(v,v)=1\right.\right\}.}$

In the English-language literature, the unit tangential bundle is also often referred to as. ${\displaystyle UTM}$

Topological properties

The unit tangential bundle is a bundle of spheres over , in particular, also a fiber bundle . The fibers are -dimensional spheres for . ${\displaystyle T^{1}M}$ ${\displaystyle M}$ ${\displaystyle (n-1)}$ ${\displaystyle n=\dim(M)}$

${\displaystyle T^{1}M}$is a -dimensional manifold. It is compact exactly when is compact. ${\displaystyle (2n-1)}$ ${\displaystyle M}$

Examples

• ${\displaystyle T^{1}S^{2}}$is diffeomorphic too . ${\displaystyle \mathbb {R} P^{3}=SO(3)}$
• ${\displaystyle T^{1}T^{2}}$ is diffeomorphic to the 3-torus.

Liouville measure

Auf is a canonical 1-form defined by ${\displaystyle T^{1}M}$ ${\displaystyle \theta }$

${\displaystyle \theta _{u}(v)=g(u,\pi _{*}v)\ \forall \ u\in T^{1}M,v\in T_{u}(T^{1}M),}$

where denotes the projection. ${\displaystyle \pi \colon T^{1}M\to M}$

The - form is a volume shape and defines a degree to which Liouville measure . ${\displaystyle (2n-1)}$ ${\displaystyle \theta \wedge (d\theta )^{n-1}}$ ${\displaystyle T^{1}M}$

${\displaystyle T^{1}M}$and the Liouville measure are invariant under the geodesic flow .

literature

• Jeffrey M. Lee: Manifolds and Differential Geometry . Graduate Studies in Mathematics Vol. 107, American Mathematical Society, Providence (2009). ISBN 978-0-8218-4815-9
• Jürgen Jost : Riemannian Geometry and Geometric Analysis , (2002) Springer-Verlag, Berlin. ISBN 3-540-42627-2
• Ralph Abraham and Jerrold E. Marsden : Foundations of Mechanics , (1978) Benjamin-Cummings, London. ISBN 0-8053-0102-X