Submanifolds Shaped Gaussian-like Geometric And Topological Properties

It plays an important role in global differential geometry to study the geometric and topological properties of Gaussian image of submanifolds. In this paper we investigate the geometric and topological properties for the Gaussian image of complete submanifolds in a space form with constant curvature. We obtain the computation formula and the estimate of lower and upper bounds for the volume of Gaussian image of submanifolds, and prove the topological sphere theorems under the pinching condition for the Gaussian image.In 1968, S. S. Chern[3] proved that the Gaussian image of complete minimal surface in R^2+P equals its total curvature times —1. In 1986, Z. H. Chen generalized the result to the case for the complete minimal surface in sphere R^2+Pand hyperbolic space. In 1990, H. Z. Li and H. W. Xu proved the computation formula for the Gaussian image of surface in a space form of constant curvature independently. In 1981, M. Gromov proved that the Betti numbers of any n-dimensional compact Riemannian manifold M satisfy: where C(n, k) is a positive constant depend on n and k only, In the first part of this paper, we study the geometric and topological properties for Gaussian image of complete oriented submanifolds in R^2+P, and give out a com-putation formula for the volume of the Gaussian image, also the lower topological bound and the upper geometric bound. Moreover we proved the topological sphere theorem under the pinching condition for the Gaussian image. Precisely, we get the result as follow. be an isometric immersion from an oriented complete Riemannian manifold M to R^n+P. Denote by g(M) the Gaussian image of M. Then (ii) If M is compact, thenIn particular, if V(g(M)) < 3C(n,p), then M is homeomorphic to Sⁿ(1). Here is the height function in the direction of z, E is the measure zero set in S^n+P-1, C_i(?_z) is the number of critical points with the index of i for Morse function (?)_z,...