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Laplacian operator is defined well on Riemannian manifold, denoted by $\Delta$. Therefore people can study PDE $\Delta f=0$ on manifold.

So is there any analogy to heat equation or wave equation on manifold? And what book is recommended for beginner to read about this field.

Thank you.

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    $\begingroup$ Heat type equation on manifold is a huge success. As a starter you can the book "Laplacian on a Riemannian manifold" by S. Rosenberg. $\endgroup$
    – user99914
    Dec 1 '14 at 11:30
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The heat and wave equations have very nice analogous equations on Riemannian manifolds $(M,g)$. If the Laplace-Beltrami operator is given by: $$ \Delta_g = \text{div}_g \,\nabla_g $$ Then the heat and wave equations, respectively, are: $$ \Delta_g u =\gamma_h\, \partial_t u $$ $$ \Delta_g u = \gamma_w\, \partial_{tt} u $$ for constants $\gamma_h,\gamma_w$. Pleasantly, their solutions can be written in terms of the spectrum of $\Delta_g$: $$ u(x,t) = \sum_i \alpha_i \exp(-\lambda_i t)\phi_i(x) $$ $$ u(x,t) = \sum_j \alpha_j \exp\left(it\sqrt{\lambda_j}\,\right)\phi_j(x) $$ where $\Delta_g \phi_\ell = -\lambda_\ell\phi_\ell$ (the Helmholtz eigenvalue equation of $M$) and $\alpha_i$ depend on the initial conditions, for the heat and wave equations respectively.

You can also compute the kernels of the PDEs, i.e. $K_t(x,y,t)$ s.t. $$ f(x,t) = \int_M K(x,y,t)f(y,0)dy $$ is a solution to the heat/wave equation, where the kernel is given by $$ K(x,y,t) = \sum_j \phi_j(x)\phi_j(y) \exp( \xi_j t) $$ with $ \xi_j = -\lambda_j $ for the heat equation and $ \xi_j=i\sqrt{\lambda_j}$ for the wave equation.

Both equations can be used to extract interesting invariants (i.e. signatures) of the manifold, which can give a great deal of information characterizing it. (The Schrodinger equation can also be treated in an almost identical manner.)

The commenter above already a mentioned a nice book, but there is also this set of notes by Canzani. The book by Berger also briefly mentions spectra.

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