---
tag: 6080X3FAD4
kind: concept
name: Regular surface
msc: [53A05]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_geometry_of_surfaces
---

## Summary

A regular surface is a subset of $\R^3$ that looks, near each of its points, like a smoothly bent piece of the plane.
Near each point it is the image of a smooth injective map from an open set of the plane, with linearly independent partial derivatives.
Regular surfaces are the setting of the classical differential geometry of surfaces: tangent planes, fundamental forms and curvatures are defined on them.

## Explanations

### L1 human

Think of the skin of a ball, or the surface of a doughnut.
If you zoom in at any point, you see something that is almost a flat sheet, bent but without folds, creases or sharp tips.
A regular surface is a shape in space with this property at every point.
The tip of a cone is a sharp point, so a cone with its tip is not a regular surface, but the sphere is.