Some Applications of the Hodge-de Rham Decomposition to Ricci Solitons

Abstract

The aim of this paper is to present a link between the Perelman potential for a compact Ricci soliton M n and the Hodge-de Rham decomposition theorem, we shall use this result to present an integral formula which enables us to establish conditions under which the Ricci soliton is trivial. Moreover, given a Ricci soliton such that its associated vector field X is a conformal vector field we show that in the compact case X is a Killing vector field, while for the non-compact case, either the soliton is Gaussian or X is a Killing vector field.Mathematics Subject Classification (2010). Primary 53C25, 53C20, 53C21; Secondary 53C65.

Some characterizations for compact almost Ricci solitons

“…Now we notice that the same result obtained in [1] for compact Ricci solitons also works for compact almost Ricci solitons. Now we notice that the same result obtained in [1] for compact Ricci solitons also works for compact almost Ricci solitons.…”

“…Finally, we obtain an integral formula for an almost Ricci soliton which is a generalization of a similar one obtained in [1] for a Ricci soliton. Theorem 4.…”

“…On the other hand, by using the Hodge-de Rham decomposition theorem (see [1]), we may decompose the vector field X over a compact oriented Riemannian manifold as a sum of the gradient of a function h and a free divergence vector field Y , i.e.…”

Integral Formulae on Quasi-Einstein Manifolds and Applications

“…Before announcing the results we point out that they are generalisations of the results due to [1,10] for Ricci solitons. Firstly, we have the following theorem.…”

“…In this section we shall show some integral formulae for a compact quasi-Einstein manifold M n , which are generalisation of the formulae obtained for Ricci solitons in [1]. Those formulae enable us to derive some rigidity results for such a class of manifolds.…”