Let $N$ and $M$ be $n$-dimensional and $n+1$-dimensional Riemannian manifolds and let $F : N \to M$ be an isometric immersion. What is the standard definition of the Laplace-Beltrami operator? $$ \Delta F = ? $$ I am interested in this particular case where the co-dimension is 1, but is there a more general definition?
Thanks!
This is known as the tension field or harmonic map Laplacian. As you'd expect, it's just $\mathrm{tr_g} (\nabla^2F)$, but we need to work out how to rigorously define this covariant second derivative of a map between manifolds.
The derivative $DF$ can be viewed as a section of the bundle $E = T^*N \otimes F^*TM$. Pulling back the Levi-Civita connection of $M$ to a connection on $F^* TM$ and then using the usual extension of connections to tensor bundles, we get a natural connection on $E$ which allows us to define $\nabla^2 F = \nabla D F$ as a section of $T^* N \otimes E = T^* N \otimes T^* N \otimes F^* TM.$ Taking the trace with respect to the metric of $N$ yields $\Delta F \in \Gamma(F^* TM)$; i.e. a vector field along $F$.