Editing Connection (vector bundle) (section)

=== Symmetric and exterior power connections ===

Since the symmetric power and exterior power of a vector bundle may be viewed naturally as subspaces of the tensor power, <math>S^k E, \Lambda^k E \subset E^{\otimes k}</math>, the definition of the tensor product connection applies in a straightforward manner to this setting. Indeed, since the symmetric and exterior algebras sit inside the [[tensor algebra]] as direct summands, and the connection <math>\nabla</math> respects this natural splitting, one can simply restrict <math>\nabla</math> to these summands. Explicitly, define the '''symmetric product connection''' by
:<math>\nabla^{\odot 2}_X(s\cdot t) = \nabla_X s \odot t + s \odot \nabla_X t</math>
and the '''exterior product connection''' by
:<math>\nabla^{\wedge 2}_X (s\wedge t) = \nabla_X s \wedge t + s\wedge \nabla_X t</math>
for all <math>s,t\in \Gamma(E), X\in \Gamma(TM)</math>. Repeated applications of these products gives induced '''symmetric power''' and '''exterior power connections''' on <math>S^k E</math> and <math>\Lambda^k E</math> respectively.