Tensor derivative
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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 .