Riemann manifolds


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:


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.

Torsion tensor

The Levi-Civita connection

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


The circle as a topology


Embeddings and immersions

Conformal maps


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.

Curvature tensor

Ricci curvature