Rough Laplacian (${\nabla }^{\ast }\nabla$)

For a metric vector bundle connection over a Riemannian manifold, the rough Laplacian is the formally nonnegative connection operatorwhere $e_i$ is a local orthonormal frame. The correction term makes the expression independent of the orthonormal frame. On a closed manifold, integration by parts gives $⟨{\nabla }^{\ast }\nabla s,s{⟩}_{L^2}=\Vert \nabla s{\Vert }_{L^2}^2$.