On the Sampson Laplacian

Abstract

In the present paper we consider the little-known Sampson operator that is strongly elliptic and self-adjoint second order differential operator acting on covariant symmetric tensors on Riemannian manifolds. First of all, we review the results on this operator. Then we consider the properties of the Sampson operator acting on one-forms and symmetric two-tensors. We study this operator using the analytical method, due to Bochner, of proving vanishing theorems for the null space of a Laplace operator admitting a Weitzenböck decomposition. Further we estimate operator's lowest eigenvalue.

A Contribution of Liouville-Type Theorems to the Geometry in the Large of Hadamard Manifolds

“…We define the non-negative scalar function f = ϕ for ϕ ∈ H(S p M). In this case, from (1) we deduce the well-known Bochner-Weitzenböck formula (see also [10,27])…”

“…In particular, the kernel of ∆ S is a finitedimensional vector space on a compact manifold (M, g). More information about the properties and applications of ∆ S can be found in the following list of articles: [10,11,26,27].…”

“…A vector field V on a complete Riemannian manifold (M, g) is called an infinitesimal harmonic transformation (see [26]) if V generates a flow which is a local one-parameter group of harmonic self-diffeomorphisms (in other words, harmonic diffeomorphisms of (M, g) onto itself). Analytic characteristic of such vector field V has the form trace g (L V ∇) = 0 for the Lie derivative L V (see [26,27]). In addition, we have proved in [27] that a vector field V is an infinitesimal harmonic transformation if and only if…”

The Sampson Laplacian on Negatively Pinched Riemannian Manifolds

“…In addition, from (1.2) we conclude that p : S p M → S p M is a symmetric endomorphism. More properties of the operator ∆ S can be found in the following papers: [24,30,31,32,33,42].…”

“…Thus, there are no negative definite Weitzenböck quadratic forms Q p of ∆ S on a Riemannian manifold with negatively 1/4-pinched sectional curvature (see [30,31,32]).…”