η-Ricci solitons and almost η-Ricci solitons on para-Sasakian manifolds

Abstract

In this paper, we study para-Sasakian manifold [Formula: see text] whose metric [Formula: see text] is an [Formula: see text]-Ricci soliton [Formula: see text] and almost [Formula: see text]-Ricci soliton. We prove that, if [Formula: see text] is an [Formula: see text]-Ricci soliton, then either [Formula: see text] is Einstein and in such a case the soliton is expanding with [Formula: see text] or it is [Formula: see text]-homothetically fixed [Formula: see text]-Einstein manifold and in such a case the soliton is shrinking with [Formula: see text]. We show the same conclusion when the para-Sasakian manifold [Formula: see text] is of [Formula: see text] and [Formula: see text] is an almost [Formula: see text]-Ricci soliton with [Formula: see text] as infinitesimal contact transformation. Finally, we prove that, if the para-Sasakian manifold [Formula: see text] of [Formula: see text] admits a gradient almost [Formula: see text]-Ricci soliton with [Formula: see text], then [Formula: see text] is Einstein. Suitable examples are constructed to justify our results.