Definition:Metric Compatible Connection

Definition
Let $\struct {M, g}$ be a Riemannian or pseudo-Riemannian manifold with or without boundary.

Let $TM$ be the tangent bundle of $M$.

Let $\map {\mathfrak{X}} M$ be the space of smooth vector fields on $M$.

Let $\nabla$ be a connection in $TM$.

Let $\innerprod \cdot \cdot$ be the Riemannian or pseudo-Riemannian scalar product.

Suppose:


 * $\forall X, Y, Z \in \map {\mathfrak{X}} M : \nabla_X \innerprod Y Z = \innerprod {\nabla_X Y} Z + \innerprod Y {\nabla_X Z}$

Then $\nabla$ is said to be compatible with $g$.