Metric linear connection

Definition

Given data

A Riemannian manifold $(M,g)$ (i.e. a differential manifold $M$ endowed with a Riemannian metric $g$).

Definition part

A metric linear connection on $M$ 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 metric connection on the tangent bundle.

An important case of a metric linear connection is the Levi-Civita connection which is the unique metric torsion-free linear connection.