  We consider a convolution-type operator on vector bundles over metric-measure
spaces. This operator extends the analogous convolution Laplacian on functions
in our earlier work to vector bundles, and is a natural extension of the graph
connection Laplacian. We prove that for Euclidean or Hermitian connections on
closed Riemannian manifolds, the spectrum of this operator and that of the
graph connection Laplacian both approximate the spectrum of the connection
Laplacian.
