---
tag: DQCD11X1VM
kind: concept
name: Shape operator
msc: [53A05]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_geometry_of_surfaces
---

## Summary

The shape operator of a surface in $\R^3$ at a point is a linear map of the tangent plane to itself.
It tells how the unit normal turns as one moves on the surface.
It is minus the differential of the Gauss map, and it is also called the Weingarten map.
Its eigenvalues are the principal curvatures, its determinant is the Gaussian curvature, and half its trace is the mean curvature.

## Explanations

### L1 human

Imagine a tiny arrow standing straight out of a surface at each point, like a hair.
Walk a little in some direction, and watch how the hair tilts.
The shape operator turns your step into the change of tilt of the hair.
On a flat floor the hair never tilts, and on a ball it tilts in the direction you walk, at the same rate in every direction.