Heat Equation on Manifold 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.
 A: 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.
