# Difference between revisions of "Gauss-Bonnet theorem for surfaces"

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The '''Gauss-Bonnet theorem''' states that the ''average'' value of Gaussian curvature over a volume-normalized compact orientable two-dimensional Riemannian manifold is proportional to the Euler characteristic of the manifold. Specifically, if <math>K</math> denotes the Gaussian curvature at point <math>p \in M</math>,then: | The '''Gauss-Bonnet theorem''' states that the ''average'' value of Gaussian curvature over a volume-normalized compact orientable two-dimensional Riemannian manifold is proportional to the Euler characteristic of the manifold. Specifically, if <math>K</math> denotes the Gaussian curvature at point <math>p \in M</math>,then: | ||

− | <math>\int_{p \in M} K dM = 2\pi\chi(M)</math> | + | {{quotation|<math>\int_{p \in M} K dM = 2\pi\chi(M)</math>}} |

Here <math>\chi(M)</math> denotes the Euler characteristic, which is a purely topological notion. | Here <math>\chi(M)</math> denotes the Euler characteristic, which is a purely topological notion. | ||

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Note that the Gauss-Bonnet theorem works ''only'' for orientable manifolds since it crucially depends on an embedding in 3-space. | Note that the Gauss-Bonnet theorem works ''only'' for orientable manifolds since it crucially depends on an embedding in 3-space. | ||

+ | When the manifold is not volume-normalized, we have the formula: | ||

+ | |||

+ | {{quotation|<math>\frac{\int_{p \in M} K dM}{\int_{p \in M} dM} = 2\pi \chi(M)</math>}} | ||

==Importance== | ==Importance== | ||

## Revision as of 14:35, 6 April 2008

Template:Curvature result for surfaces

*This result relates information on curvature to information on topology of a manifold*

*This result is about manifolds in dimension:*2

## Contents

## Statement

The **Gauss-Bonnet theorem** states that the *average* value of Gaussian curvature over a volume-normalized compact orientable two-dimensional Riemannian manifold is proportional to the Euler characteristic of the manifold. Specifically, if denotes the Gaussian curvature at point ,then:

Here denotes the Euler characteristic, which is a purely topological notion.

Note that the Gauss-Bonnet theorem works *only* for orientable manifolds since it crucially depends on an embedding in 3-space.

When the manifold is not volume-normalized, we have the formula:

## Importance

### As a tool for area computation

The Gauss-Bonnet theorem can be used to compute areas of regions in surfaces having constant curvature. For instance, it can be used to obtain a formula for the area of a geodesic triangle on the two-dimensional sphere, embedded in three-space.

### As a topological control on the Gaussian curvature

The Gauss-Bonnet theorem tells us that given a manifold, we cannot give it both a metric with everywhere positive Gaussian curvature, and a metric with everywhere negative curvature.

It also tells us that if we fix the total volume of the manifold, then there is only one possible value for the curvature, if it is constant.

## Relation with other results

### Riemann-Roch theorem

`Further information: Riemann-Roch theorem`

The Gauss-Bonnet theorem, which is a tool for computation of the Euler characteristic, can be viewed as a special case of the Hirzebruch-Riemann-Roch theorem, which is a more general version of the Riemann-Roch theorem for curves. Let's see what this means:

- A result of Riemann-Roch type is one that computes an Euler characteristic-type quantity (which is a sum (or difference) of dimensions of certain vector spaces), in terms of certain characteristic classes
- The Gauss-Bonnet theorem computes the Euler characteristic as an integral of a curvature over a region. It is of Riemann-Roch type because the integral of curvature can be viewed as arising from certain characteristic classes.