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If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the Laplacian into normal and tangential components w.r.t the submanifold?

I have found that section 14.2 of "Fundamentals of Differential geometry" by Serge Lang has material on decomposing the covariant Laplacian of a function $\nabla^2 f$ w.r.t. a submanifold, but I am wondering whether there is a more physicist-friendly reference, or one that explicitly extends the formalism to the Laplacian acting on tensors.

Many thanks,

A physicist

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  • $\begingroup$ Do you really want to consider the Laplacian acting on the metric? Or is there a better example of what you want to do? $\endgroup$ – Muphrid Jun 14 '14 at 3:56
  • $\begingroup$ I actually want the Laplacian to be acting on an arbitray tensor $h_{\mu\nu}$. $\endgroup$ – user42587 Jun 14 '14 at 4:06

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