Metric linear connection

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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:

X g(Y,Z) = g(\nabla_XY,Z) + g(Y,\nabla_XZ)

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: Induced connection on submanifold of metric connection is metric