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 $M$, with a smoothly varying metric structure on eac hfibre of $E$ over $M$.

Definition part

A metric connection on $(M,g)$ is a connection $\nabla$ on the vector bundle $E$ over $M$ satisfying the following condition:

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

Here $X$ is a vector field (viz a section of $TM$) and $Y$ and $Z$ are sections of $E$.

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

We are in particular interested in metric linear connections, which are metric connections over the tangent bundle. Of these, a very special one is the Levi-Civita connection, which is the only torsion-free metric linear connection.