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Differential Geometry of Manifolds

2020 №51

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Stepanov S. E. Tsyganok I. I.

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

DOI
10.5922/0321-4796-2020-51-13
Pages
116-122
Riemannian manifold conformal Killing — Yano tensor sectional curvature vanishing theorem

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.

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