This lives as an element of: the space of all linear connections, which in turn sits inside the space of all -bilinear maps
More generally, we can also look at a pseudo-Riemannian manifold, or a manifold with a pseudo-Riemannian metric: a smoothly varying nondegenerate (not necessarily positive definite) symmetric bilinear form in each tangent space.
A Levi-Civita connection on is a linear connection on satisfying the following two conditions:
Definition by formula
The formula for the Levi-Civita connection requires us to use the fact that is nondegenerate. So, instead of directly specifying for vector fields and , the formula specifies for vector fields, as follows:
The term Levi-Civita connection is sometimes also used for the connection induced on any tensor product involving the tangent and cotangent bundle, using the rule for tensor product of connections and the dual connection.
The Levi-Civita connection is unique
Further information: Levi-Civita connection exists and is unique
The proof combines the metric condition, the torsion-free condition, and the nondegeneracy of .
Note that the nondegeneracy of is very important, otherwise knowing the value of the inner product for any three vectors may not necessarily help us in computing the value of
Further information: Christoffel symbols of a connection
The Levi-Civita connection on a manifold is a map . This means that at any point , it gives a map , which roughly differentiates one tangent vector along another.
Let form a basis for the tangent space . Then, the Christoffel symbol is the component along of the vector .
The Christoffel symbols thus give an explicit description of the Levi-Civita connection. Namely, the Levi-Civita connection can be expressed using the Christoffel symbols.