Levi-Civita Connection in Local Orthonormal Frame

Theorem

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

Let $U \subseteq M$ be an open subset of $M$.

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

Let $\tuple {E_i}$ be a smooth local orthonormal frame.

Let $c^k_{ij} : U \to \R$ be smooth functions such that:

$\sqbrk {E_i, E_j} = c^k_{ij} E_k$

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

Let $g$ be a Riemannian or pseudo-Riemannian metric, which locally reads:

$g = g_{ij} \rd x^i \otimes \rd x^j$

where $\paren {g_{ij}}$ is a matrix of smooth functions.

Let $g^{ij}$ be the inverse of $\paren {g_{ij}}$.

Suppose $\Gamma^k_{ij}$ are the connection coefficients of $\nabla$.

Then in the frame $\tuple {E_i}$ we have:

$\Gamma^k_{ij} = \dfrac 1 2 \paren {c^k_{ij} - c^j_{ik} - c^i_{jk} }$

Further research is required in order to fill out the details. In particular: Coordinate chart involvement needs clearer description You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more. 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 {{Research}} from the code.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. 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 {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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