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 $TM$.

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.

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

Suppose:

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

where $\nabla$ on the left hand side denotes the connection in the $0$-tensor field, i.e.:

$\nabla_X \innerprod Y Z = \map X {\innerprod Y Z}$

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

Sources

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