On The Unit Sphere And The Hyperbolic Space Of Submanifolds In Some Of The Results

This paper includes two parts. In chapter one, We established a topological sphere theorem from the viewpoint of submanifold geometry for even-dimensional submanifolds Mn in the unit sphere, in terms of Ricci curvature and the mean curvature. This result is a generalization of Vlachos's theorem from odd-dimension to even-dimension. In chapter two, by investigating hypersurfaces M" in the hyperbolic space Hn+1(-1) with constant scalar curvature and with two distinct principal curvatures, We got the classification theorem and some rigidity results.Now, we can state our main results as follows.Theorem A Let Mn be a compact oriented submanifold of the unit sphere Sn+k with mean curvature vector H. If n is even and the Ricci curvature satisfiesthen Mn is homeomorphic to Sn.Theorem B Let Mn be an n-dimensioal hypersurface in Hn+1(-l)(n ≥ 3) with constant scalar curvaturen(n -1 )R and two distinct principal curvatures. Then the following hold. (i) If the multiplicities of the two principal curvatures are all constant and greater than one, then Mn is isoparametric and isometric to the Riemannian product Hm(c1) x Sn-m(c2) for some 2 ≤ m ≤ n -2, where c1 < 0 and c2 > 0 are constants and satisfying 1/c1 + 1/c2 =-1;(ii) There exists infinitely many hypersurfaces with constant scalar curvature and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n - 1;(iii) If Mn is complete and the multiplicity of one of the principal curvatures is n - 1. Then R ≥ -1. Moreover, if we assume that R > -(n-2)/n, (R≠ 0), and the norm square Sof the second fundamental form of Mn satisfies(n- l)(nR + n-2) n-2n-2 nR + n-2'then Mn is isometric to either the Riemannian product H1( — nR/(nR + n — 2)) x Sn~1(nR/(n-2)) for R > 0 or the Riemannian product Hn-1(ni2/(n-2)) xS1(-nR/(nR+ n - 2)) for R < 0.(iv) If Mn is complete and having constant scalar curvature n(n — 1)R(R > 0) and with two distinct principal curvatures, one of which is simple. Assume that the norm square 5 of the second fundamental form of Mn satisfies n-2- n-2 ni? + n-2'then Mn is isometric to H*( - nR/(nR + n - 2)) x Sn~1(nR/(n - 2)).