This article is cited in 1 paper

Infinitesimal Harmonic Transformations and Ricci Solitons

S. E. Stepanov^a, I. G. Shandra^a, V. N. Shelepova<sup>b

a</sup> Finance Academy under the Government of the Russian Federation, Moscow
^b Vladimir State Humanitarian University

Abstract: A Ricci soliton on a smooth manifold $M$ is a triple $(g_0,\xi,\lambda)$, where $g_0$ is a complete Riemannian metric, $\xi$ a vector field, and $\lambda$ a constant such that the Ricci tensor $\mathrm{Ric}_0$ of $g_0$ satisfies the equation $-2\mathrm{Ric}_0=L_\xi g_0+2\lambda g_0$. In the paper, we study the geometry of Ricci solitons in dependence of the properties of the vector field $\xi$. In particular, we prove that this vector field is a harmonic transformation.

Keywords: Riemannian manifold, infinitesimal harmonic transformation, Ricci soliton.

UDC: 514.764.2

Received: 25.08.2009