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 $\mathrm{\nabla }$ on $M$ satisfying the following condition:

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

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.

Facts

Given a Riemannian manifold $M$, a submanifold $N$, and a metric linear connection on $M$, the induced linear connection on the submanifold $N$ is also a metric connection.

For full proof, refer: Restriction of metric linear connection is metric