Tangent bundle

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

Definition

The tangent bundle of $M$ is the set

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

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

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 diffeomorphism 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.

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

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

1. 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 $\pi(x,v)=x$.

2. 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.

3. 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 .)