# Difference between revisions of "Main Page"

From Diffgeom

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=Contents= | =Contents= | ||

− | *[Lie group] | + | ==Basics== |

+ | |||

+ | *[[Topological manifold]] | ||

+ | *[[Differential manifold]] | ||

+ | *[[Riemannian manifold]] | ||

+ | *[[Connection]] | ||

+ | *[[Lie group]] | ||

+ | |||

+ | ===Properties of connections=== | ||

+ | |||

+ | *[[Linear connection]] | ||

+ | *[[Torsion of a linear connection]] | ||

+ | *[[Torsion-free linear connection]] | ||

+ | |||

+ | ===Basic kinds of metrics=== | ||

+ | |||

+ | *[[Riemannian metric]] | ||

+ | *[[Homogeneous metric]] | ||

+ | *[[Locally homogeneous metric]] | ||

+ | |||

+ | ==Classification of differential manifolds according to the structure group== | ||

+ | |||

+ | *[[Pseudo-Riemannian manifold]] | ||

+ | *[[Almost Hermitian manifold]] | ||

+ | *[[Almost complex manifold]] | ||

+ | *[[Almost symplectic manifold]] | ||

+ | *[[Lorentzian manifold]] | ||

+ | *[[Oriented manifold]] | ||

+ | |||

+ | ==Curvatures== | ||

+ | |||

+ | *[[Riemann curvature tensor]] | ||

+ | *[[Ricci curvature tensor]] | ||

+ | *[[Scalar curvature]] | ||

+ | *[[Sectional curvature]] | ||

+ | *[[Total scalar curvature]] | ||

+ | *[[Total absolute scalar curvature]] | ||

+ | *[[Average scalar curvature]] | ||

+ | *[[Mean curvature]] | ||

+ | *[[Average mean curvature]] | ||

+ | |||

+ | ==Classification of metrics according to curvatures== | ||

+ | |||

+ | *[[Flat metric]] | ||

+ | *[[Ricci-flat metric]] | ||

+ | *[[Einstein metric]] | ||

+ | *[[Constant-curvature metric]] | ||

+ | *[[Constant-scalar curvature metric]] | ||

+ | *[[Zero-scalar curvature metric]] | ||

+ | |||

+ | ==Special scalar functions on Riemannian manifolds== | ||

+ | |||

+ | *[[Injectivity radius of a manifold]] | ||

+ | *[[Isoperimetric constant]] | ||

+ | *[[Pinching constant]] | ||

+ | |||

+ | ==Various theorems== | ||

+ | |||

+ | ===Results on Ricci curvature=== | ||

+ | |||

+ | *[[Bonnet-Myers theorem]] | ||

+ | *[[Schoen-Yau theorem]] | ||

+ | *[[Zhu's theorem]] | ||

+ | *[[Splitting theorem]] | ||

+ | |||

+ | ===Results on sectional curvature=== | ||

+ | |||

+ | *[[Cheeger-Gromoll conjecture]] | ||

+ | *[[Cohn-Vossen theorem]] | ||

+ | *[[Gromoll-Meyer theorem]] | ||

+ | *[[Cartan-Hadamard theorem]] | ||

+ | *[[Bieberbach theorem]] | ||

+ | *[[Finiteness theorem]] | ||

+ | *[[Schur's theorem]] | ||

+ | *[[Synge's theorem]] | ||

+ | |||

+ | ===Results relating curvature to topology=== | ||

+ | |||

+ | *[[Bieberbach theorem]] | ||

+ | *[[Bonnet-Myers theorem]] | ||

+ | *[[Cartan-Hadamard theorem]] | ||

+ | *[[Gauss-Bonnet theorem]] | ||

+ | *[[Synge's theorem]] | ||

+ | *[[Preissmann's theorem]] | ||

+ | |||

+ | |||

+ | |||

+ | *[[Splitting theorem]] | ||

+ | |||

+ | ==Special topics== | ||

+ | |||

+ | *[[Minimal manifold]] | ||

+ | *[[Characteristic class]] | ||

+ | *[[Chern form]] | ||

+ | *[[Chern-Weil theorem]] |

## Revision as of 20:13, 7 July 2007

This is Diffgeom (the Differential Geometry wiki).

Much of it is patterned on the Group Properties wiki.

It is currently in a very preliminary stage and may contain a number of factual errors. It is meant only for private use.

It may never become sturdy enough for public use!

## Contents

# Contents

## Basics

### Properties of connections

### Basic kinds of metrics

## Classification of differential manifolds according to the structure group

- Pseudo-Riemannian manifold
- Almost Hermitian manifold
- Almost complex manifold
- Almost symplectic manifold
- Lorentzian manifold
- Oriented manifold

## Curvatures

- Riemann curvature tensor
- Ricci curvature tensor
- Scalar curvature
- Sectional curvature
- Total scalar curvature
- Total absolute scalar curvature
- Average scalar curvature
- Mean curvature
- Average mean curvature

## Classification of metrics according to curvatures

- Flat metric
- Ricci-flat metric
- Einstein metric
- Constant-curvature metric
- Constant-scalar curvature metric
- Zero-scalar curvature metric

## Special scalar functions on Riemannian manifolds

## Various theorems

### Results on Ricci curvature

### Results on sectional curvature

- Cheeger-Gromoll conjecture
- Cohn-Vossen theorem
- Gromoll-Meyer theorem
- Cartan-Hadamard theorem
- Bieberbach theorem
- Finiteness theorem
- Schur's theorem
- Synge's theorem

### Results relating curvature to topology

- Bieberbach theorem
- Bonnet-Myers theorem
- Cartan-Hadamard theorem
- Gauss-Bonnet theorem
- Synge's theorem
- Preissmann's theorem