Covariant Derivative of Smooth Vector Field along Smooth Vector Field in Smooth Local Frame
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Theorem
Let $M$ be a smooth manifold with or without boundary.
Let $TM$ be the tangent bundle of $M$.
Let $\nabla$ be a connection in $TM$.
Let $U \subseteq M$ be an open subset.
Suppose $\tuple {E_i}$ is a smooth local frame over $U$.
Let $\set {\Gamma^k_{ij}}$ be the connection coefficients of $\nabla$ with respect to $\tuple {E_i}$.
Let $\map {\mathfrak X} U$ be the space of smooth vector fields on $U$.
Let $X, Y \in \map {\mathfrak X} U$ be smooth vector fields.
Suppose $X$ and $Y$ read as $X = X^i E_i$ and $Y = Y^i E_i$ with respect to $\tuple {E_i}$.
Then:
- $\nabla_X Y = \paren {\map X {Y^k} + X^i Y^j \Gamma^k_{ij}} E_k$
Proof
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Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Connections in the Tangent Bundle