Constant Angle Surfaces In H~3×R

There is a long history on theory of Riemannian Geometry and great results have researched so far. With the development of modern society, Geometry is widely used in mathematics, physics, mechanics etc. However, recently the research work has expanded to hypersurfaces of the product spaces for arbitrary dimension, a lot of work has been given the characterization of constant angle surfaces in H2×R, S2×R, S3×R etc.In this thesis, this paper discusses constant angle surfaces in H3x R. The results and characterizations given in this paper extend and improve the corresponding ones in the literatures.In Chapter1, it introduces the fundamental concepts and theory of H3×R.First, it defines constant angle surfaces in H3x R, next, for tangent vector fields X, Y, Z to M and normal vector fields N, it gives the following Gauss equation, Codazzi equation and Ricci equation: Gauss equation:(R(X, Y)Z)T=R(X, Y)Z+Ah(X,Z)Y-Ah(Y,z)X, Codazzi equation:(R(X, Y)Z)(?)=((?)(?)/x)(Y,Z)-((?)(?)/Yh)(X, Z), Ricci equation:(R(X, Y)N)(?)=R(?)(X,Y)N+h(ANX, Y)-h(X, ANY).In Chapter2, it mainly discusses the problems of minimal surfaces in H3x R, the main results are as follows:If M is a constant angle surface in H3×R, with parallel mean curvature vector H, then the shape operators with respect to ξ, ηare respectively:In Chapter3, it mainly discusses the characterization of constant angle surfaces in H6×R with parallel mean curvature vector, the main results are as follows:A surface M is constant angle surface with parallel mean curvature vector in H3×R,if and only of the immcrsion is,at most up to isometries of H3×R,locally given by F(x,y)=(1/2(y2+1)eεcosθ+1/2e-εcosθ,1/2(y2-1)eεcosθ+1/2e-εcosθ,yeεcosθ,0,xsinθ) whereε=±1,constant angleθ∈(0,π/2).