Hitchin–Thorpe inequality and Kaehler metrics for compact almost Ricci soliton

Abstract

The purpose of this paper is to prove a Hitchin-Thorpe inequality for a fourdimensional compact almost Ricci soliton. Moreover, we prove that under a suitable integral condition, a four-dimensional compact almost Ricci soliton is isometric to standard sphere. Finally, we prove that under a simple condition, a four-dimensional compact Ricci soliton with harmonic self-dual part of Weyl tensor is either isometric to a standard sphere S 4 or is Kaehler-Einstein.

Ricci almost solitons on semi‐Riemannian warped products

“…Motivated by these relations with geometric flows, Ricci solitons and Einstein manifolds we are interested in investigating the geometry of such manifolds and we consider problems such as when a Ricci almost soliton becomes a Ricci soliton or even an Einstein manifold. In [1], [3], [4], [6], [21], [24], [37] and [40] the authors proved that under certain geometric constraints a Ricci almost soliton becomes a Ricci soliton or an Einstein manifold carrying a conformal field.…”

On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons

“…In the Kähler case, it is known that any compact Kähler gradient Ricci soliton of real dimension 4 with the natural orientation satisfies the inequality 2χ(M ) + 3τ (M ) ≥ 0 (see [29]; this result was generalized to Kähler almost Ricci solitons in [2]).…”

Four‐dimensional gradient almost Ricci solitons with harmonic Weyl curvature

“…In [21] the authors presented some introductory properties of gradient almost Ricci solitons, and in particular, classified all those which are Einstein metrics. In [4] Hitchin–Thorpe inequality was shown for almost Ricci solitons under some condition. Compact gradient almost Ricci solitons were studied in [3], so that if they have either constant scalar curvature or an associated conformal vector field, then they are isometric to a Euclidean sphere; see also [2].…”