A connection Laplacian is the second-order operator 
associated with a vector
bundle connection on a Riemannian manifold.
The adjoint uses the Riemannian
metric and the fiber inner product. For a unitary connection 
on a trivial Hermitian bundle, this is an elliptic operator
on vector-valued sections. For the trivial connection
on scalar functions, it reduces to the positive Laplace-Beltrami operator.
On a smooth compact manifold with boundary, its Dirichlet-to-Neumann operator can be specified by the energy pairing
 |
where 
with boundary values 
.
