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Structures And Satisfaction Point Bijection Neighborhood Obvious Manifold Function Differntiable Function Vector Field Types Of Morphisms Isomorphic Groups Derivative Disjoint Union Cone In Mathbb R Direct Product Of Algebras Tangent Space Vector Bundle Section Of A Fiber Bundle Fibre

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Tangent Bundle

Let be a differentiable manifold. Let the tangent bundle of be(as a set) the disjoint union $\coprod_{m\in M}T_mM$ of all the tangent spaces to , i.e., the set of pairs

$\displaystyle \{(m,x)\vert m\in M, x\in T_mM\}.$

This naturally has a manifold structure, given as follows. For $M=\mathbb{R}^n$ , $T\mathbb{R}^n$ is obviously isomorphic to $\mathbb{R}^{2n}$ , and is thus obviously a manifold. By the definition of a differentiable manifold, for any $m\in M$ , there is a neighborhood

and a diffeomorphism $\varphi :\mathbb{R}^n\to U$ . Since this map is a diffeomorphism, its derivative is an isomorphism at all points. Thus $T\varphi :T\mathbb{R}^n=\mathbb{R}^{2n}\to TU$ is bijective, which endows

with a natural structure of a differentiable manifold. Since the transition maps for

are differentiable, they are for

as well, and

is a differentiable manifold. In fact, the projection $\pi:TM\to M$ forgetting the tangent vector and remembering the point, is a vector bundle. A vector field on

is simply a section of this bundle.

The tangent bundle is functorial in the obvious sense: If $f:M\to N$ is differentiable, we get a map $Tf:TM\to TN$ , defined by on the base, and its derivative on the fibers.

1	Manifold
1	Point
1	Differntiable Function
1	Neighborhood
1	Obvious
1	Structures And Satisfaction
1	Function
1	Bijection
2	Types Of Morphisms
2	Derivative
2	Isomorphic Groups
2	Vector Field
3	Tangent Space
3	Cone In Mathbb R
3	Direct Product Of Algebras
3	Disjoint Union
4	Fibre
5	Vector Bundle
6	Section Of A Fiber Bundle