Definition:Levi-Civita 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 $\nabla$ be a connection on $TM$.

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

Then $\nabla$ is called Levi-Civita connection.

Source of Name

This entry was named for Tullio Levi-Civita.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Symmetric Connections