Covariant derivative
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and $\Gamma E$ the space of smooth sections. A {\it covariant derivative} | and $\Gamma E$ the space of smooth sections. A {\it covariant derivative} | ||
on $E$ is a bilinear map $\nabla : \Gamma TM \times \Gamma E \to \Gamma E$, $(X,s)\mapsto\nabla_Xs$, | on $E$ is a bilinear map $\nabla : \Gamma TM \times \Gamma E \to \Gamma E$, $(X,s)\mapsto\nabla_Xs$, | ||
− | which is a [[tensor]] (linear over $C^\infty(M)$) in the first argument and | + | which is a [[Tensor|tensor]] (linear over $C^\infty(M)$) in the first argument and |
a {\it derivation} in the second argument: | a {\it derivation} in the second argument: | ||
{{equation|$\begin{matrix}\nabla_{(fX)}s &=& f\nabla_Xs\, \\ \nabla_{X}(fs) &=& (Xf)s + f\nabla_Xs \,,\end{matrix}$|1}} | {{equation|$\begin{matrix}\nabla_{(fX)}s &=& f\nabla_Xs\, \\ \nabla_{X}(fs) &=& (Xf)s + f\nabla_Xs \,,\end{matrix}$|1}} |
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1 Definition
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and the space of smooth sections. A {\it covariant derivative} on is a bilinear map , , which is a tensor (linear over ) in the first argument and a {\it derivation} in the second argument:
Tex syntax errorand a section of , and where
is the ordinary derivative of the function in the direction of . By these properties, is defined locally and even pointwise regarding the first argument: For any we may define where is any (local) vector field with .
A covariant derivative has all properties of the ordinary directional derivative for functions with exception of the commutativity: For functions we have with , where is
a local parametrization ofTex syntax errorand
its -th partial derivative. Instead, for covariant derivatives, the commutator (with ) is nonzero in general, but it is only a tensor (rather than a differential operator), the curvature tensor.
Covariant derivatives are not tensors since they are derivations in the second argument, but two covariant derivative on differ by a tensor: defines a tensor .