Almost Quasi-Yamabe Solitons In Space Forms

In this paper,we mainly investigate almost Yamabe solitons and almost quasi-Yamabe solitons on submanifolds in space forms.By using the special vector field in space forms and the basic formulas and equations on submanifolds,we discussed the soliton fields of almost Yamabe solitons and almost quasi-Yamabe solitons on submanifolds in space forms,and further obtained that any almost Yamabe solitons and almost quasi-Yamabe solitons on a hypersurface in space forms are totally um-bilical and quasi-umbilical,respectively.The main results obtained in this paper include the following three parts.1.The gradient almost Yamabe solitons and gradient almost quasi-Yamabe solitons under compact condition are studied.By using the divergence theorem and the partial integral formula,we obtained that their Ricci mean value are non-negative,and further proved that solitons are trival if and only if their Ricci mean value are vanishing.2.By finding a special vector field in space forms and using the basic formulas and equations on submanifolds,we obtained a sufficient and necessary condition for submanifolds in space forms to be almost Yamabe solitons,meanwhile,the soliton fields of the almost Yamabe solitons are closed conformal vector fields,we further proved that any almost Yamabe solitons on a hypersurface in space forms are totally umbilical.3.By using ideas and methods of the second part,we obtained a sufficient and necessary condition for submanifolds in space forms to be almost quasi-Yamabe solitons,at the same time,the soliton fields of the almost quasi-Yamabe solitons are Torse-forming vector fields.Through the research for the eigenvalues of the shape operator of a hypersurface,we further proved that any almost quasi-Yamabe solitons on a hypersurface in space forms are quasi-umbilical,and contained in a rotation hypersurface.