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 .
definition
Let it be a Riemannian manifold and its tangential bundle . The unit tangent bundle is
In the English-language literature, the unit tangential bundle is also often referred to as.
Topological properties
The unit tangential bundle is a bundle of spheres over , in particular, also a fiber bundle . The fibers are -dimensional spheres for .
is a -dimensional manifold. It is compact exactly when is compact.
Examples
- is diffeomorphic too .
- is diffeomorphic to the 3-torus.
Liouville measure
Auf is a canonical 1-form defined by
where denotes the projection.
The - form is a volume shape and defines a degree to which Liouville measure .
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