INVESTIGATIONS ON A RIEMANNIAN MANIFOLD WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION AND GRADIENT SOLITONS

This article carries out the investigation of a three-dimensional Riemannian manifold N 3 endowed with a semi-symmetric type non-metric connection. Firstly, we construct a non-trivial example to prove the existence of a semi-symmetric type non-metric connection on N 3 . It is established that a N 3 with the semi- symmetric type non-metric connection, whose metric is a gradient Ricci soliton, is a manifold of constant sectional curvature with respect to the semi-symmetric type non-metric connection. Moreover, we prove that if the Riemannian metric of N 3 with the semi-symmetric type non-metric connection is a gradient Yamabe soliton, then either N 3 is a manifold of constant scalar curvature or the gradient Yamabe soliton is trivial with respect to the semi-symmetric type non-metric connection. We also characterize the manifold N 3 with a semi-symmetric type non-metric connection whose metrics are Einstein solitons and in-quasi Einstein solitons of gradient type, respectively.