📂Geometry

Tangent Bundles on Differentiable Manifolds

Definition¹

Let’s call a $M$ a $n$-dimensional differentiable manifold. Let’s denote the tangent space at point $p \in M$ as $T_{p}M$. The tangent bundle $TM$ of $M$ is defined as follows.

$$ \begin{align*} TM &:= \bigsqcup \limits_{p \in M } T_{p}M \\ &= \bigcup_{p \in M} \left\{ p \right\} \times T_{p}M \\ &= \left\{ (p, v) : p \in M, v \in T_{p}M \right\} \end{align*} $$

Here, $\bigsqcup$ is a disjoint union.

Explanation

By definition, the tangent bundle is a set of all ordered pairs of all points on the differentiable manifold $M$ and all tangent vectors at those points. As can be seen in the disjoint union document, it’s possible to consider a natural mapping between $(p,v)$ and $v$, effectively treating them as the same thing, thus sometimes $\bigsqcup$ is replaced by $\bigcup$.

$$ TM := \bigcup_{p \in M} T_{p}M $$

If $M$ is a $n$-dimensional differentiable manifold, then $TM$ itself becomes a $2n$-dimensional differentiable manifold again.