Levi-Civita Connection in Terms of Vector Fields
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Theorem
Let $\struct {M, g}$ be a Riemannian or pseudo-Riemannian manifold with or without boundary.
Let $\map {\mathfrak X} M$ be the space of smooth vector fields on $M$.
Let $\sqbrk {\cdot, \cdot}$ be the Lie bracket.
Let $\innerprod \cdot \cdot$ be the Riemannian or pseudo-Riemannian scalar product.
Suppose $\nabla$ is the Levi-Civita connection of $\struct {M, g}$.
Then:
- $\forall X, Y, Z \in \map {\mathfrak X} M : \innerprod {\nabla_X Y} Z = \dfrac 1 2 \paren {X \innerprod Y Z + Y \innerprod Z X - Z \innerprod X Y - \innerprod Y {\sqbrk {X, Z} } - \innerprod Z {\sqbrk {Y, X} } + \innerprod X {\sqbrk {Z, Y} } }$
Proof
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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