Editing Connection (vector bundle) (section)

=== Endomorphism connection ===

Finally, one may define the induced connection <math>\nabla^{\operatorname{End}{E}}</math> on the vector bundle of endomorphisms <math>\operatorname{End}(E) = E^* \otimes E</math>, the '''endomorphism connection'''. This is simply the tensor product connection of the dual connection <math>\nabla^*</math> on <math>E^*</math> and <math>\nabla</math> on <math>E</math>. If <math>s\in \Gamma(E)</math> and <math>u\in \Gamma(\operatorname{End}(E))</math>, so that the composition <math>u(s) \in \Gamma(E)</math> also, then the following product rule holds for the endomorphism connection:
:<math>\nabla_X(u(s)) = \nabla_X^{\operatorname{End}(E)} (u) (s) + u(\nabla_X (s)).</math>

By reversing this equation, it is possible to define the endomorphism connection as the unique connection satisfying

:<math> \nabla_X^{\operatorname{End}(E)} (u) (s) = \nabla_X(u(s)) - u(\nabla_X(s))</math>

for any <math>u,s,X</math>, thus avoiding the need to first define the dual connection and tensor product connection.