On the Tachibana numbers of closed manifolds with pinched negative sectional curvature

Abstract

Conformal Killing form is a natural generalization of con­formal Killing vector field. These forms were exten­si­vely studied by many geometricians. These considerations we­re motivated by existence of various applications for the­se forms. The vector space of conformal Killing p-forms on an n-dimensional closed Riemannian mani­fold M has a finite dimension na­med the Tachibana number. These numbers are conformal scalar invariant of M and satisfy the duality theorem: . In the present article we prove two vanishing theorems. According to the first theorem, there are no nonzero Tachi­bana numbers on an n-dimensional closed Rie­mannian manifold with pinched negative sectional curva­ture such that for some pinching con­stant . From the second theorem we conc­lude that there are no nonzero Tachibana numbers on a three-dimensional closed Riemannian manifold with ne­gative sectional curvature.

Keywords

• riemannian manifold
• conformal killing — yano tensor
• sectional curvature
• vanishing theorem