On the Killing property of the defining vector field for an almost Yamabe soliton

Abstract

In this paper, we first investigate almost Yamabe solitons on compact Riemannian manifolds without boundary of dimension greater than or equal to two. We provide some sufficient conditions for which the defining conformal vector field associated to a compact almost Yamabe soliton is a Killing vector field. We then study almost Yamabe solitons on complete, non-compact Riemannian manifolds. We prove the Killing property of the defining conformal vector field associated to a complete, non-compact almost Yamabe soliton under certain conditions when the dimension is strictly greater than two.

v3: 10 pages. Added some results for non-compact case also. So, the title, abstract and Introduction (i.e Section 1) are modified accordingly. v2: 9 Pages. Revised the conjecture, added Lemma 2.3, and some changes in Section 4 and References