Tensor derivative

Revision as of 13:16, 15 March 2013

1 Definition

Let ${{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 $M$, 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:

$(\nabla_XT)s = \nabla_X(Ts) - T(\nabla_Xs)$(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:

$\nabla_X(Ts) = (\nabla_XT)s + T(\nabla_Xs)$( )

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

$(R(X,Y)T)s = R(X,Y)Ts - T(R(X,Y)s)$(2)

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

2 References