Compatibility
Let
be a smooth Riemannian manifold with metric . A connection on is said to be compatible with the metric if for every pair of vector fields and on , and every vector ,
Remark
It is equivalent to say that
. However, this is not a good notation because it does not make sense to apply the affine connection on a -tensor. It is meaningful once we define that where is the dual connection of .
Levi-Civita Connection
An affine connection defined on
is called a Levi-Civita connection if it is compatible with the metric and it is torsion free.
Fundamental Theorem of Riemannian Geometry
Every Riemannian manifold admits a unique Levi-Civita connection. In particular, the unique Levi-Civita connection has the following coordinate expression: