How is the tangent bundle of a manifold defined set-theoretically?

Q: How is the tangent bundle of a manifold defined set-theoretically?

The tangent bundle $TM$ of a manifold $M$ is defined as the disjoint union of all tangent spaces $T_pM$ for every point $p$ in $M$. Symbolically, $TM = \bigsqcup_{p \in M} T_pM$.

The tangent bundle

TM

of a manifold

M

is defined as the disjoint union of all tangent spaces

T_pM

for every point

p

M

. Symbolically,

TM = \bigsqcup_{p \in M} T_pM

Conditions

$M$ is a smooth manifold.
$T_pM$ is the tangent space at point $p$ .

Reasoning, step by step

Identify all points $p$ in the manifold $M$ .
For each $p$ , consider the tangent space $T_pM$ .
Take the disjoint union of all these tangent spaces.
Denote this union as $TM$ .

Example

The video writes: 'TM = $\bigsqcup_{p \in M} T$ _pM.'

Common misconceptions

Thinking the tangent bundle is just the manifold itself.
Confusing the tangent bundle with the cotangent bundle.

Watch the explanation

BilibiliRiemannian Geometry_Part 2_Riemannian metrics_John Lee_Introduction to Riemannian Manifolds p01 11_Riemannian metrics

5:37 – 6:00Watch this moment ↗

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