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Mat. Zametki, 2012, Volume 92, Issue 3, Pages 459–462 (Mi mz9010)  

This article is cited in 2 scientific papers (total in 2 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

DOI: https://doi.org/10.4213/mzm9010

Full text: PDF file (409 kB)
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English version:
Mathematical Notes, 2012, 92:3, 422–425

Bibliographic databases:

UDC: 514.764.2
Received: 28.03.2011

Citation: S. E. Stepanov, I. G. Shandra, “New Characteristics of Infinitesimal Isometry and Ricci Solitons”, Mat. Zametki, 92:3 (2012), 459–462; Math. Notes, 92:3 (2012), 422–425

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. E. Stepanov, I. I. Tsyganok, “Harmonic Transforms of Complete Riemannian Manifolds”, Math. Notes, 100:3 (2016), 465–471  mathnet  crossref  crossref  mathscinet  isi  elib
    2. I. A. Aleksandrova, S. E. Stepanov, I. I. Tsyganok, “Ot garmonicheskikh otobrazhenii k potokam Richchi na osnove tekhniki Bokhnera”, Materialy mezhdunarodnoi konferentsii “Geometricheskie metody v teorii upravleniya i matematicheskoi fizike”, posvyaschennoi 70-letiyu S.L. Atanasyana, 70-letiyu I.S. Krasilschika, 70-letiyu A.V. Samokhina, 80-letiyu V.T. Fomenko. Ryazanskii gosudarstvennyi universitet im. S.A. Esenina, Ryazan, 25–28 sentyabrya 2018 g. Chast 2, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 169, VINITI RAN, M., 2019, 75–87  mathnet  crossref
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