= Rough Laplacian
{title2=$\nabla^*\nabla$}
For a metric vector bundle connection over a <Riemannian manifold>, the rough Laplacian is the formally nonnegative connection operator
$$
\nabla^*\nabla=-\sum_i\bigl(\nabla_{e_i}\nabla_{e_i}-\nabla_{\nabla_{e_i}e_i}\bigr),
$$
where $e_i$ is a local orthonormal frame. The correction term makes the expression independent of the orthonormal frame. On a <closed manifold>, integration by parts gives $\langle\nabla^*\nabla s,s\rangle_{L^2}=\|\nabla s\|_{L^2}^2$.
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