---
tag: WY6K3P4YCG
kind: concept
name: Tangent bundle
msc: [58A05, 57R22]
specializes: [CX4M8K61AB]
sources:
  - url: https://en.wikipedia.org/wiki/Tangent_bundle
same_as:
  wikidata: Q746550
  wikipedia: Tangent bundle
---

## Summary

The tangent bundle of a smooth manifold collects all tangent vectors at all points into one space.
A point of the tangent bundle is a pair: a point of the manifold and a velocity vector at that point.
The tangent bundle is itself a smooth manifold of twice the dimension, and it is the basic example of a vector bundle.
Vector fields are its sections, and the derivative of a smooth map is a map between tangent bundles.

## Explanations

### L1 human

At each point of a sphere there is a tangent plane, the flat plane that just touches the sphere there.
The tangent bundle puts all these planes together, one for each point.
To specify a point of it, you give a place on the sphere and an arrow lying in the tangent plane at that place.
For the plane $\R^2$ itself, the tangent bundle is just $\R^2 \times \R^2$: a position and an arrow.