Tangent bundle - Knowlpedia

Let $M$ be a smooth manifold of dimension $n$ .

The tangent bundle of $M$ is the set

TM = \bigsqcup_{p\in M} T_pM,

the disjoint union of all tangent spaces of $M$ , equipped with the projection map

\pi: TM \to M,\qquad \pi(v)=p \ \text{for } v\in T_pM.

There is a canonical smooth manifold structure on $TM$ characterized as follows: for any smooth chart $(U,\varphi)$ on $M$ with $\varphi:U\to \mathbb{R}^n$ , the induced map

\pi^{-1}(U) \longrightarrow \varphi(U)\times \mathbb{R}^n

sending a tangent vector $v\in T_pM$ to $(\varphi(p), (v^1,\dots,v^n))$ in the coordinate basis $\left.\frac{\partial}{\partial x^i}\right|_p$ is a smooth coordinate chart onto an open subset of $\mathbb{R}^{2n}$ . These charts are compatible across a smooth atlas and make $\pi:TM\to M$ into a smooth map.

Consequence

With this structure, $TM$ is a rank- $n$ vector bundle over $M$ : each fiber $\pi^{-1}(p)=T_pM$ is a vector space, and the local identifications above give smooth local trivializations.

Sections and vector fields

A smooth section of $\pi:TM\to M$ is the same thing as a vector field on $M$ : it assigns to each $p\in M$ a tangent vector $X(p)\in T_pM$ smoothly in $p$ .

Functoriality

Any smooth map $f:M\to N$ induces a map on tangent bundles

df:TM\to TN

defined fiberwise by the differential (pushforward) $d f_p:T_pM\to T_{f(p)}N$ . This “bundle map” covers $f$ in the sense that $\pi_N\circ df = f\circ \pi_M$ .

Examples

Euclidean space is trivial. For $M=\mathbb{R}^n$ , the tangent bundle is canonically diffeomorphic to a product: $T\mathbb{R}^n \cong \mathbb{R}^n \times \mathbb{R}^n,$ with $\pi(x,v)=x$ .

The circle is also trivial. The tangent bundle of $S^1$ is trivial: $TS^1 \cong S^1\times \mathbb{R}.$ Concretely, the unit tangent vector field $X(\cos t,\sin t)=(-\sin t,\cos t)$ provides a global framing.

Products. For smooth manifolds $M$ and $N$ , there is a canonical identification $T(M\times N) \cong TM \times TN$ compatible with the projections to $M\times N$ . Under this identification, a tangent vector at $(p,q)$ is a pair $(v,w)$ with $v\in T_pM$ and $w\in T_qN$ .

(For comparison, the dual bundle of $TM$ is the cotangent bundle.)