Tensor product of connections
Definition
Suppose are vector bundles over a differential manifold
. Suppose
is a connection on
and
is a connection on
. The tensor product
is defined as the unique connection on
such that the following is satisfied for all sections
of
respectively:
.
Properties
Well-definedness
Further information: Tensor product of connections is well-defined
It is not completely clear from the definition that the tensor product of connections is well-defined. What needs to be shown is that the definition given above for pure tensor products of sections can be extended to all sections consistently, while maintaining the property of being a connection.
Associativity
Further information: Tensor product of connections is associative upto natural isomorphism
Suppose are vector bundles over a differential manifold
, with connections
respectively. Then, under the natural isomorphism:
,
the connections and
get identified.
Commutativity
Further information: Tensor product of connections is commutative upto natural isomorphism
Suppose are vector bundles over a differential manifold
, with connections
respectively. Then, under the natural isomorphism:
the connections and
get identified.
Distributivity with direct sum
Further information: Distributivity relation between direct sum and tensor product of connections
Suppose are vector bundles over a differential manifold
, with connections
respectively. Then, under the natural isomorphism:
we have an identification between and
. Here,
is the direct sum of connections.
An analogous distributivity law identifies and
.
Commutes with dual connection operation
Further information: Dual connection to tensor product equals tensor product of dual connections
Relation with other interpretations of connection
Tensor product of module structures
Further information: Connection is module structure over connection algebra, Tensor product of connections corresponds to tensor product of modules over connection algebra
A connection on a vector bundle can be thought of as an interpretation of its global sections as a module over the connection algebra. With this interpretation, a tensor product of connections corresponds to the tensor product of these modules over the connection algebra.
Tensor product of connections viewed as splittings
Fill this in later
Facts
Formula for Riemann curvature tensor
Further information: Formula for curvature of tensor product of connections