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

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