# Riemann manifolds

## Introduction

### Metric tensors

A metric tensor assigns a bilinear form to each point on the manifold.

We can then take two vectors in the tangent space and return a scalar.

### Riemann manifolds and pseudo-Riemann manifolds

#### Riemann manifolds

Metric is positive definite.

#### Pseudo-Riemann manifolds

The metric isn’t necessarily positive definite.

### Metric tensor field

metric tensor field assigns a metric tensor to each point. metric tensor is defined on the tangent bundle. so we have metric on each tangent bundle, but the metric can change thoughout the manifold

### Length of paths in Riemann manifolds

We can work out the length of a path through a Riemann manifold.

The geodesic is the shortest such path.

The Riemann metric between two points is the length of the geodesic.

## Connections on Riemann manifolds

### Metric compatibility

If we have two vectors in the tangent space of a manifold with a metric tensor, we can get a scalar:

$$v^iu^jg_{ij}$$

#### Transported metric

If we transport two vectors along a connection, we have the metric at the new point.

#### Metric preserving connections

If the connection preserves the metric, then the connection is metric compatible.

### The Levi-Civita connection

For any metric tensor there is only one connection which preserves the metric and is torsion free.

## Sort

### Geodesics

How do we have straight line on a curve? eg going round equator, but not going via uk.

Take start direction and find tangent vectors. geodesic is where tangent vectors stay parallel.