The differential operators in the bundle of symmetric tensors on a Riemannian manifold

Abstract

Differential operators: the gradient grad and the divergence div are defined and examined in the bundles of symmetric tensors on a Riemannian manifold. For the second order operator div grad which appears to be elliptic and a manifold with boundary a system of natural boundary conditions is constructed and investigated. There are 2k+1 conditions in the bundle Sk of symmetric tensors of degree k. This is in contrast to the bundle of skewsymmetric forms where (for analogous differential operators) there are always four such conditions independently of the degree of forms (i.e. independently of k). All the 2k+1 conditions are investigated in detail. In particular, it is proved that each of them is self-adjoint and elliptic. Such the ellipticity of a given boundary condition has an essential significance for the existing of an orthonormal basis in L2 consisting of smooth sections that are the eigenvalues of the operator and satisfy the boundary condition. Some special cases, e.g. k = 1 or the the cases that the boundary is umbilical or totally geodesic are also discussed.