This article is cited in 5 papers

New Characteristics of Infinitesimal Isometry and Ricci Solitons

S. E. Stepanov, I. G. Shandra

Financial University under the Government of the Russian Federation

Abstract: We prove that a vector field $X$ on a compact Riemannian manifold $(M,g)$ with Levi-Cività connection $\nabla$ is an infinitesimal isometry if and only if it satisfies the system of differential equations: $\operatorname{trace}_g(L_X\nabla)=0$, $\operatorname{trace}_g(L_X\operatorname{Ric})=0$, where $L_X$ is the Lie derivative in the direction of $X$ and $\operatorname{Ric}$ is the Ricci tensor. It follows from the second assertion that the Ricci soliton on a compact manifold $M$ is trivial if its vector field $X$ satisfies one of the following two conditions: $\operatorname{trace}_g(L_X\operatorname{Ric})\le 0$ or $\operatorname{trace}_g(L_X \operatorname{Ric})\ge 0$.

Keywords: compact Riemannian manifold, infinitesimal isometry, Levi–Cività connection, vector field, Ricci soliton, Ricci tensor, local harmonic transformation.

UDC: 514.764.2

Received: 28.03.2011

DOI: 10.4213/mzm9010

 English version: Mathematical Notes, 2012, 92:3, 422–425

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