---
tag: F45VQ47GPV
kind: concept
name: Smooth manifold
msc: [58A05, 57R55]
specializes: [QQQ9B1DK4P]
sources:
  - url: https://en.wikipedia.org/wiki/Differentiable_manifold
---

## Summary

A smooth manifold is a space that looks like a piece of $\R^n$ near each point, glued together so that calculus makes sense on it.
The gluing maps between overlapping coordinate charts are required to be infinitely differentiable.
This lets one speak of smooth functions, tangent vectors, vector fields and differential forms on curved spaces such as spheres and tori.
Smooth manifolds are the basic objects of differential geometry and differential topology.

## Explanations

### L1 human

The surface of the Earth is curved, but a small region of it can be drawn on a flat map.
An atlas is a collection of such maps that together cover the whole Earth, with overlaps.
A smooth manifold is a space with an atlas in which passing from one map to an overlapping one is a smooth change of coordinates.
The circle is a one-dimensional example: near each point it looks like a piece of a line, and the angle coordinate works as a chart away from one point.