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 $TM$.

The validity of the material on this page is questionable. In particular: Does $\nabla$ not have to be a connection on $M$ according to the definition? $TM$ is another manifold. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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