Metric connection

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

Definition

Given data

A differential manifold $M$.

A metric bundle $E$ over $M$ (viz, a vector bundle with a smoothly varying metric structure $g$ on each fibre 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: $X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)$

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 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.