Tensor derivative
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Let $M$ be a smooth manifold and $E,E'$ vector bundles over $M$, both equipped with a [[Covariant derivative|covariant derivative]] $\nabla$. | 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 [[ | + | Then the vector bundle $\textup{Hom}(E,E')$ of bundle homomorphisms (sometimes called [[Tensor|tensors]]) inherits another ''covariant derivative'' |
$\nabla : \Gamma TM \times \Gamma\textup{Hom}(E,E') \to \Gamma\textup{Hom}(E,E')$ | $\nabla : \Gamma TM \times \Gamma\textup{Hom}(E,E') \to \Gamma\textup{Hom}(E,E')$ | ||
called the ''tensor derivative'' which is defined as follows: | called the ''tensor derivative'' which is defined as follows: |
Revision as of 11:56, 15 March 2013
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1 Definition
Let be a smooth manifold and vector bundles over , both equipped with a covariant derivative . Then the vector bundle of bundle homomorphisms (sometimes called tensors) inherits another covariant derivative called the tensor derivative which is defined as follows:
(1)
for any and . The definition is made such that the application of tensors (sections in ) to sections in satisfies the Leibniz product rule:
The corresponding curvature tensors of the bundles , and are related similarly:
(2)
for any and .