This article is cited in 11 papers

Theorems of existence and non-existence of conformal Killing forms

S. E. Stepanov^a, I. I. Tsyganok<sup>b

a</sup> Chair of Mathematics, Financial University at the Government of the Russian Federation, 49–55 Leningradskii Ave., Moscow, 125993 Russia
^b Chair of Probability theory and Mathematical Statistics, Financial University at the Government of the Russian Federation, 49–55 Leningradskii Ave., Moscow, 125993 Russia

Abstract: On an $n$-dimensional compact, orientable, connected Riemannian manifold, we consider the curvature operator acting on the space of covariant traceless symmetric $2$-tensors. We prove that, if the curvature operator is negative, the manifold admits no nonzero conformal Killing $p$-forms for $p=1,2,\dots,n-1$. On the other hand, we prove that the dimension of the vector space of conformal Killing $p$-forms on an $n$-dimensional compact simply-connected conformally flat Riemannian manifold $(M, g)$ is not zero.

Keywords: Riemannian manifold, curvature operator, conformal Killing forms, vanishing theorem, existence theorem.

UDC: 514.764

Received: 30.03.2013

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2014, 58:10, 46–51

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