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

where *T*_{x}*M* 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 *T*_{x}*M* 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 *TM* ⊕ *E* 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 ... on M (say, a Riemannian metric), there is no similar lift into the cotangent

**bundle**...

... characteristic of the n-sphere is Thus, there is no non-vanishing section of the

**tangent bundle**of even spheres, so the

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**bundles**... Since the

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**tangent Bundle**s Of) Riemann Surfaces

... 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 ... let M=HΓ be 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 ...

... See also Connection (principal

**bundle**) Let M be a surface embedded in E3 ... unit vector n is defined at each point of the surface and hence a determinant can be defined on

**tangent**vectors v and w at that point using the usual ... An ordered basis or frame v, w in the

**tangent**space is said to be oriented if det(v, w) is positive ...

**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 bundle**(TM,πT ... The second

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