Real Hypersurfaces In CP~1×CP~1

Theories of submanifolds in Riemannian manifolds are important parts of Riemannian geometries. In particular, the di?erential geometry of hypersurfaces in complex spaces has rich contents. The study of real hypersurfaces in complex space forms has been one of active fields of study over the past decade and the results are very ample. Naturally, the study of real hypersurfaces in product space of complex space forms is an interesting topic.In the present dissertation, we mainly study real hypersurfaces in product K¨ahler manifold CP1×CP1. We introduce related knowledges and results in three chapters.In chapter 1, we detailedly introduce the background of study and some related knowledges.In chapter 2, we mainly introduce important properties of two hypersurfaces, namely,properties of umbilical hypersurface and hypersurface with parellel shape operator. Then,we give a special hypersurface, namely, Berger sphere in CP1×CP1.In chapter 3, we give two theorems for Hopf hypersurfaces. When θ is constant, we prove the existence of Hopf hypersurface. When θ is not constant, we demonstrate the property of Hopf hypersurface.