Editing Connection (vector bundle) (section)

== Induced connections ==

Given a vector bundle <math>E\to M</math>, there are many associated bundles to <math>E</math> which may be constructed, for example the dual vector bundle <math>E^*</math>, tensor powers <math>E^{\otimes k}</math>, symmetric and antisymmetric tensor powers <math>S^k E, \Lambda^k E</math>, and the direct sums <math>E^{\oplus k}</math>. A connection on <math>E</math> induces a connection on any one of these associated bundles. The ease of passing between connections on associated bundles is more elegantly captured by the theory of [[Connection (principal bundle)|principal bundle connections]], but here we present some of the basic induced connections.

=== 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>.

=== Tensor product connection ===
Given <math>\nabla^E, \nabla^F</math> connections on two vector bundles <math>E, F\to M</math>, define the '''tensor product connection''' by the formula
:<math> (\nabla^E \otimes \nabla^F)_X(s\otimes t) = \nabla_X^E (s) \otimes t + s\otimes \nabla_X^F (t). </math>
Here we have <math>s\in \Gamma(E), t\in \Gamma(F), X\in \Gamma(TM)</math>. Notice again this is the natural way of combining <math>\nabla^E, \nabla^F</math> to enforce the product rule for the tensor product connection. By repeated application of the above construction applied to the tensor product <math>E^{\otimes k} = (E^{\otimes (k-1)}) \otimes E</math>, one also obtains the '''tensor power connection''' on <math>E^{\otimes k}</math> for any <math>k\ge 1</math> and vector bundle <math>E</math>.

=== Direct sum connection ===

The '''direct sum connection''' is defined by

:<math> (\nabla^E \oplus \nabla^F)_X (s\oplus t) = \nabla_X^E (s) \oplus \nabla_X^F (t),</math>
where <math>s\oplus t\in \Gamma(E\oplus F)</math>.

=== 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.

=== 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.

=== Any associated bundle ===

{{See also|Connection (principal bundle)}}

Given a vector bundle <math>E</math> of rank <math>r</math>, and any representation <math>\rho: \mathrm{GL}(r,\mathbb{K}) \to G</math> into a linear group <math>G\subset \mathrm{GL}(V)</math>, there is an induced connection on the associated vector bundle <math>F = E\times_\rho V</math>. This theory is most succinctly captured by passing to the principal bundle connection on the [[frame bundle]] of <math>E</math> and using the theory of principal bundles. Each of the above examples can be seen as special cases of this construction: the dual bundle corresponds to the inverse transpose (or inverse adjoint) representation, the tensor product to the tensor product representation, the direct sum to the direct sum representation, and so on.