Levi-Civita Connection in Terms of Vector Fields

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

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

Let $\sqbrk {\cdot, \cdot}$ be the Lie bracket.

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

Suppose $\nabla$ is the Levi-Civita connection of $\struct {M, g}$.

Then:


 * $\ds \forall X, Y, Z \in \map {\mathfrak{X}} M : \innerprod {\nabla_X Y} Z = \frac 1 2 \paren {X \innerprod Y Z + Y \innerprod Z X - Z \innerprod X Y - \innerprod Y {\sqbrk {X, Z} } - \innerprod Z {\sqbrk {Y, X} } + \innerprod X {\sqbrk {Z, Y} } }$