Almost Schouten solitons and contact geometry

Abstract

The current article is about almost Schouten solitons and gradient Schouten solitons on contact geometry. At first, we demonstrate that if a compact $K$-contact manifold admits an almost Schouten soliton, then the soliton is shrinking and the manifold is an Einstein manifold. Moreover, we show that if a $K$-contact manifold admits a gradient Schouten soliton, then the manifold becomes an Einstein manifold. Next, we investigate almost Schouten solitons and gradient Schouten solitons on $(k,\mu)$-contact manifolds. Finally, we show that if a complete $H$-contact manifold $M^{2n+1}$ satisfying certain restriction on the scalar curvature and the soliton function admits an almost Schouten soliton whose potential vector field $V$ is collinear with $\zeta$, then $M^{2n+1}$ is compact Einstein and Sasakian.