The Geometric Characteristics Of The Clifford Torus. Unit Sphere

In this thesis,we mainly study the second and first pinching problems in a sphere,and get some geometric characterizations of Clifford torus in a sphere.We first study the second pinching problem of closed minimal hypersurface and prove the following result.Let M be an n-dimensional closed minimal hypersurface in the unit sphereSⁿ⁺¹(1). If there exists at most one simple principal curvature at each point p∈M,andthen S≡n ,and M is the Clifford torus S^k(?)×S^n-k(?).We generalize the result to the case of closed hypersurface with constant mean curvature.Let M be the n-dimensional closed hypersurface with constant mean curvatureofa unit sphere Sⁿ⁺¹ (1), 0≤H < D(n) ,D(n) is a constant only rely on n. If there exists at most one simple principal curvature at each point p∈M,andthen S≡β(n, H) ,i.e.M is a Clifford hypersurfacewhereWe study the first pinching problem for compact submanifolds in an unit sphere and get the following result.Let M be an n-dimensional compact submanifold with parallel mean curvature vector and flat normal bundle in S^n+p(1). If S≤α(n,H),then M is either thetotally umbilical sphere Sⁿ (1/(?)),the Clifford isoparametric hypersurfaceS^n-1(1/(?))×S¹(λ(n,H)/(?)) in a totally geodesic sphere Sⁿ⁺¹(1), orthe Clifford torus S¹(r₁)×S¹(r₂)in S³(r) with constant mean curvature H₀, where...