We will discuss some natural linear differential operators for
different geometric structures.
For a Riemannian manifold of dimension n, an interesting family consist
of operators of form $S^*S$, where $S^*$ is the operator formally adjoint to $S$
and where $S$ is the the gradient in the sense of Stein and Weiss, i.e., $S$
is an $O(n)$-irreducible summand of the covariant derivative.
We will discuss the ellipticity and the boundary properties such
operators. In particular, we will discuss natural boundary conditions
for the elliptic operators and the ellipticity of these conditions at
the boundary. One of the consequences of such the ellipticity for a
given boundary condition is the existence of a basis for $L^2$ composed of
smooth sections that are the eigenvectors of the operator and satisfy
the boundary condition.
We will also discuss the Laplace type operators of form $\mathop{\rm div} \mathop{\rm grad}$ acting
in tensor bundles on a Riemannian or symplectic manifold. Here the
operator grad is a natural generalization of the classic gradient
operator acting on vector fields. The negative divergence $-\mathop{\rm div}$ is the
operator formally adjoint to $\mathop{\rm grad}$. The second order operator $-\mathop{\rm div} \mathop{\rm grad}$
relates to the Lichnerowicz Laplacian which acts on tensors (forms) of
any symmetry. The relation involves the curvature.
We will also mention the problem of restriction of differential
operators (so the Stein-Weiss gradients in particular) to submanifolds
or to the leaves of a foliation.