Almost Yamabe Solitons In Riemannian Manifolds

In this thesis,we mainly study isometrical immersion of gradient almost Yam-abe solitons and r-almost Newton-Yamabe solitons into a Riemannian manifold.By using Hopf's maximum principles and the basic equations of the submanifold,we ob-tain the sufficient conditions for the gradient almost Yamabe solitons and r-almost Newton-Yamabe solitons to be totally geodesic,or totally umbilical ones.1.We study the gradient almost Yamabe soliton Mn immersed into a Rieman-nian manifold Mn+q.By using Hopf's maximum principles and the basic equations of the submanifold,we discuss whether Mn is a minimal submanifold,and obtain the sufficient conditions for gradient almost Yamabe solitons to be totally geodesic,or totally umbilical ones.For a compact minimal gradient almost Yamabe soli-ton(Mn,g,f,?)in a unit Euclidean sphere Sn+1,we prove that Mn isometrics to Euclidean sphere if its scalar curvature S?n(n-2).2.We consider the r-almost Newton-Yamabe soliton Mn immersed into a Rie-mannian manifold Mn+1.Under the condition that Mn has a bounded second fundamental form,or the r-th Newton transformation Pr is bounded from above in the sense of quadratic forms and the potential function is nonnegative,we dis-cuss whether Mn is a minimal submanifold and obtain the sufficient conditions for r-almost Newton-Yamabe solitons to be totally geodesic,or totally umbilical ones.3.We investigate the 1-almost Newton-Yamabe soliton Mn immersed into a locally symmetric Einstein manifold Mn+1.By using ideas and methods of the second part,continuing to discuss whether Mn is a minimal submanifold,and we obtain the sufficient conditions for 1-almost Newton-Yamabe solitons to be totally geodesic,or totally umbilical ones.