Fundamental Theorem of Riemannian Geometry

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

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

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

Suppose $\nabla$ is compatible with $g$ and symmetric.

Then $\nabla$ is unique and equal to Levi-Civita connection.