Editing Connection (vector bundle) (section)

=== Dual connection ===
Given <math>\nabla</math> a connection on <math>E</math>, the induced '''dual connection''' <math>\nabla^*</math> on <math>E^*</math> is defined implicitly by
:<math> d(\langle \xi, s \rangle)(X) = \langle \nabla_X^* \xi, s \rangle + \langle \xi, \nabla_X s \rangle.</math>
Here <math>X\in \Gamma(TM)</math> is a smooth vector field, <math>s\in \Gamma(E)</math> is a section of <math>E</math>, and <math>\xi \in \Gamma(E^*)</math> a section of the dual bundle, and <math>\langle \cdot , \cdot \rangle</math> the natural pairing between a vector space and its dual (occurring on each fibre between <math>E</math> and <math>E^*</math>), i.e., <math>\langle \xi , s \rangle:= \xi(s)</math>. Notice that this definition is essentially enforcing that <math>\nabla^*</math> be the connection on <math>E^*</math> so that a natural [[product rule]] is satisfied for pairing <math> \langle \cdot , \cdot \rangle</math>.