Curvature of direct sum of connections equals direct sum of curvatures
From Diffgeom
Statement
Suppose are vector bundles over a differential manifold
. Suppose
are connections on
respectively. Let
denote the Riemann curvature tensor of
and
denote that Riemann curvature tensor of
. Then if
is the direct sum of vector bundles,
is the direct sum of connections, and
is its Riemann curvature tensor, we have:
.