Certain results on \(K\)-contact and \((k, \mu )\)-contact manifolds
A Ricci soliton is a generalization of an Einstein metric defined on a Riemannian manifold \((M,g)\) by the condition \(L_Vg+2Ric+2\lambda g=0\), where \(V\) is a complete vector field on \(M\) and \(\lambda\) is a constant. According to the sign of \(\lambda\), the soliton is said to be shrinking, steady, or expanding, respectively. If the vector field \(V\) is the gradient of a potential function \(-f\), then \(g\) is called a gradient Ricci soliton. A Ricci soliton on a compact manifold is a gradient Ricci soliton. In dimension 2 and 3, a Ricci soliton on a compact manifold has constant curvature. According to a previous result, a compact Einstein K-contact manifold is Sasakian. In the present work, this is generalized by considering a complete K-contact metric as a Ricci soliton. It is proved that, if the metric of a complete K-contact manifold \((M,g)\) is a gradient soliton, then \(g\) is a shrinking soliton and \((M,g)\) is compact Einstein and Sasakian. For the nongradient case, the soliton vector field is a Jacobi vector field along the geodesics of the Reeb vector field. Other results show that among complete and simply connected K-contact manifolds only the unit sphere admits a non-Killing holomorphically planar conformal vector field and, if a \((k,\mu)\)-contact manifold admits a non-zero holomorphically planar conformal vector field, then it is either Sasakian or locally isometric to \(E^3\) of \(E^{n+1}\times S^n(4)\).
holomorphically planar conformal vector field
scientific article; zbMATH DE number 5562593