New York Journal of Mathematics
Volume 1 (1994-1995) 10-25


O. Gil-Medrano and A. Montesinos Amilibia

About a Decomposition of the Space of Symmetric Tensors of Compact Support on a Riemannian Manifold

Published: July 13, 1994
Keywords: manifold of Riemannian metrics, elliptic operators on non-compact manifolds, manifolds of maps
Subject: Primary: 58D15, 58D17. Secondary: 58G25

Let M be a noncompact manifold and let Γc(S2(M)) (respectively Γc(T1(M))) be the LF space of 2-covariant symmetric tensor fields (resp. 1-forms) on M, with compact support. Given any Riemannian metric g on M, the first-order differential operator δ*:Γc(T1(M))→Γc(S2(M)) can be defined by
δ*ω = 2 symm∇ω,
where ∇ denotes the Levi-Civita connection of g.

The aim of this paper is to prove that the subspace Im δ* is closed and to show several examples of Riemannian manifolds for which

Γc(S2(M)) ≠ Im δ* ⊕ (Imδ*),
where orthogonal is taken with respect to the usual inner product defined by the metric.

Author information

Departamento de Geometría y Topología. Facultad de Matemáticas. Universidad de Valencia. 46100 Burjasot, Valencia. SPAIN.