Metric connection

This lives as an element of: the space of all connections, which in turn sits inside the space of all $\mathbb {R}$-bilinear maps $\Gamma (TM)\times \Gamma (E)\to \Gamma (E)$

Definition

Given data

A Riemmanian manifold $(M,g)$ (here, $M$ is a differential manifold and $g$ is the additional structure of a Riemannian metric on it).

A vector bundle $E$ over $M$. More generally, we can also look at a pseudo-Riemannian manifold, or a manifold with a smoothly varying nondegenerate (not necessarily positive definite) symmetric positive definite bilinear form in each tangent space.

Definition part

A metric connection on $(M,g)$ is a linear connection $\nabla$ on $M$ satisfying the following condition:

$Xg(Y,Z)=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z)$

In other words, it is a connection such that the dual connection on the cotangent bundle is the same as the connection obtained by the natural isomorphism between the tangent and cotangent bundle (induced by the metric).