# Main Page

From Diffgeom

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