Tensor derivative

Revision as of 13:16, 15 March 2013 (view source) Diarmuid Crowley (Talk | contribs) ← Older edit		Revision as of 10:51, 15 May 2013 (view source) Diarmuid Crowley (Talk | contribs) m Newer edit →
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	$\nabla : \Gamma TM \times \Gamma\textup{Hom}(E,E') \to \Gamma\textup{Hom}(E,E')$		$\nabla : \Gamma TM \times \Gamma\textup{Hom}(E,E') \to \Gamma\textup{Hom}(E,E')$
	called the ''tensor derivative'' which is defined as follows:		called the ''tensor derivative'' which is defined as follows:
−	{{equation|$(\nabla_XT)s = \nabla_X(Ts) - T(\nabla_Xs)$|1}}	+	\begin{equation} (\nabla_XT)s = \nabla_X(Ts) - T(\nabla_Xs) \end{equation}
	for any $X \in \Gamma TM$ and $s\in \Gamma E$.		for any $X \in \Gamma TM$ and $s\in \Gamma E$.
	The definition is made such that the application of		The definition is made such that the application of
	tensors (sections in $\textup{Hom}(E,E')$) to sections in $E$ satisfies the Leibniz product rule:		tensors (sections in $\textup{Hom}(E,E')$) to sections in $E$ satisfies the Leibniz product rule:
−	{{equation|$\nabla_X(Ts) = (\nabla_XT)s + T(\nabla_Xs)$| }}	+	\begin{equation} \nabla_X(Ts) = (\nabla_XT)s + T(\nabla_Xs) \end{equation}
	The corresponding [[Curvature tensor and second derivative|curvature tensors]] of the bundles $E$, $E'$ and $\textup{Hom}(E,E')$ are related similarly:		The corresponding [[Curvature tensor and second derivative|curvature tensors]] of the bundles $E$, $E'$ and $\textup{Hom}(E,E')$ are related similarly:
−	{{equation|$(R(X,Y)T)s = R(X,Y)Ts - T(R(X,Y)s)$|2}}	+	\begin{equation} (R(X,Y)T)s = R(X,Y)Ts - T(R(X,Y)s) \end{equation}
	for any $X,Y \in \Gamma TM$ and $s\in \Gamma E$.		for any $X,Y \in \Gamma TM$ and $s\in \Gamma E$.
	</wikitex>		</wikitex>
−
	== References ==		== References ==
	{{#RefList:}}		{{#RefList:}}
	[[Category:Definitions]]		[[Category:Definitions]]
		+	[[Category:Connections and curvature]]

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Revision as of 10:51, 15 May 2013

The user responsible for this page is Jost Eschenburg. No other user may edit this page at present.

This page has not been refereed. The information given here might be incomplete or provisional.

1 Definition

Let

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${{Authors|Jost Eschenburg}}{{Stub}} ==Definition == <wikitex>; Let $M$ be a smooth manifold and $E,E'$ vector bundles over $M$, both equipped with a [[Covariant derivative|covariant derivative]] $\nabla$. Then the vector bundle $\textup{Hom}(E,E')$ of bundle homomorphisms (sometimes called [[Tensor|tensors]]) inherits another ''covariant derivative'' $\nabla : \Gamma TM \times \Gamma\textup{Hom}(E,E') \to \Gamma\textup{Hom}(E,E')$ called the ''tensor derivative'' which is defined as follows: {{equation|$(\nabla_XT)s = \nabla_X(Ts) - T(\nabla_Xs)$|1}} for any $X \in \Gamma TM$ and $s\in \Gamma E$. The definition is made such that the application of tensors (sections in $\textup{Hom}(E,E')$) to sections in $E$ satisfies the Leibniz product rule: {{equation|$\nabla_X(Ts) = (\nabla_XT)s + T(\nabla_Xs)$| }} The corresponding [[Curvature tensor and second derivative|curvature tensors]] of the bundles $E$, $E'$ and $\textup{Hom}(E,E')$ are related similarly: {{equation|$(R(X,Y)T)s = R(X,Y)Ts - T(R(X,Y)s)$|2}} for any $X,Y \in \Gamma TM$ and $s\in \Gamma E$. </wikitex> == References == {{#RefList:}} [[Category:Definitions]]M$ be a smooth manifold and $E,E'$ vector bundles over

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$M$, both equipped with a covariant derivative $\nabla$.

Then the vector bundle $\textup{Hom}(E,E')$ of bundle homomorphisms (sometimes called tensors) inherits another covariant derivative $\nabla : \Gamma TM \times \Gamma\textup{Hom}(E,E') \to \Gamma\textup{Hom}(E,E')$ called the tensor derivative which is defined as follows:

for any $X \in \Gamma TM$ and $s\in \Gamma E$. The definition is made such that the application of tensors (sections in $\textup{Hom}(E,E')$) to sections in $E$ satisfies the Leibniz product rule:

The corresponding curvature tensors of the bundles $E$, $E'$ and $\textup{Hom}(E,E')$ are related similarly:

for any $X,Y \in \Gamma TM$ and $s\in \Gamma E$.

2 References