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.
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.
