This lives as an element of: the space of all connections, which in turn sits inside the space of all -bilinear maps
A metric connection on is a connection on the vector bundle over satisfying the following condition:
Here is a vector field (viz a section of ) and and are sections of .
In other words, it is a connection such that the dual connection on the dual bundle to is the same as the connection obtained by the natural isomorphism between and its dual (induced by the metric).
We are in particular interested in metric linear connections, which are metric connections over the tangent bundle for a Riemannian manifold (viz, the tangent bundle is endowed with a Riemannian metric). Of these, a very special one is the Levi-Civita connection, which is the only torsion-free metric linear connection.