Tangent Bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint union of the tangent spaces of M. That is,

where TxM denotes the tangent space to M at the point x. So, an element of TM can be thought of as a pair (x, v), where x is a point in M and v is a tangent vector to M at x. There is a natural projection

defined by π(x, v) = x. This projection maps each tangent space TxM to the single point x.

The tangent bundle to a manifold is the prototypical example of a vector bundle (a fiber bundle whose fibers are vector spaces). A section of TM is a vector field on M, and the dual bundle to TM is the cotangent bundle, which is the disjoint union of the cotangent spaces of M. By definition, a manifold M is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold M is framed if and only if the tangent bundle TM is stably trivial, meaning that for some trivial bundle E the Whitney sum TME is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for n=1,3,7 (by results of Bott-Milnor and Kervaire).

Read more about Tangent Bundle:  Role, Topology and Smooth Structure, Examples, Vector Fields, Higher-order Tangent Bundles, Canonical Vector Field On Tangent Bundle, Lifts

Other articles related to "tangent bundle, tangent, bundle, bundles":

Tangent Bundle - Lifts
... For example, if c is a curve in M, then c' (the tangent of c) is a curve in TM ... a Riemannian metric), there is no similar lift into the cotangent bundle ...
Riemannian Connection On A Surface - Orthonormal Frame Bundle
... See also Connection (principal bundle) Let M be a surface embedded in E3 ... and hence a determinant can be defined on tangent vectors v and w at that point using the usual scalar triple product on E3 (itself a determinant) ... An ordered basis or frame v, w in the tangent space is said to be oriented if det(v, w) is positive ...
Anosov Diffeomorphism - Anosov Flow On (tangent Bundles Of) Riemann Surfaces
... As an example, this section develops the case of the Anosov flow on the tangent bundle of a Riemann surface of negative curvature ... This flow can be understood in terms of the flow on the tangent bundle of the Poincare half-plane model of hyperbolic geometry ... a Riemann surface of negative curvature, and let T1M be the tangent bundle of unit-length vectors on the manifold M, and let T1H be the tangent bundle of unit-length vectors on H ...
Double Tangent Bundle
... In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent ... The second tangent bundle arises in the study of connections and second order ordinary differential equations, i.e ... and it is not to be confused with the second order jet bundle ...
Euler Class - Examples - Spheres
... of the n-sphere is Thus, there is no non-vanishing section of the tangent bundle of even spheres, so the tangent bundle is not trivial -- i.e ... we can use the fact that the Euler class of a Whitney sum of two bundles is just the cup product of the Euler class of the two bundles to see that there are no ... Since the tangent bundle of the sphere is stably trivial but not trivial, all other characteristic classes vanish on it, and the Euler class is the only ordinary cohomology class that detects non-triviality of ...

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