# Metric linear connection

From Diffgeom

## Definition

### Given data

A Riemannian manifold (i.e. a differential manifold endowed with a Riemannian metric ).

### Definition part

A **metric linear connection** on is a linear connection on satisfying the following condition:

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 , a submanifold , and a metric linear connection on , the induced linear connection on the submanifold is also a metric connection.

*For full proof, refer: Induced connection on submanifold of metric connection is metric*