Moebius Hypersurfaces With Constant Sectional Curvature In Sⁿ⁺¹

This paper consists of four sections. The first section is the introduction. In the second section, we will give the fundamental theory for Mobius geometry of submanifolds, the Mobius invariant and the structure equations. and In the third section, we will give the first part of proof of our theorem. In the last section, we will give the proof of our classification theorem. The details are these.In the second section, we gave the fundamental theory for Mobius geometry of sub-manifolds, and gave the introduction of the Mobius form(?), Mobius second fundamental form B, Blaschke tensor A and the structure equations.In the third section, we proofed the first part of our theorem.Lemma Let x:M→S(n+1)(n≥3) be an n-dimensional umbilic-free hypersurface. If x is a constant sectional hypersurface with respect to the Mobius metric g, then for any point p∈M, the multiplicities of the distinct eigenvalues of the Mobius second fundamental form B and the Blaschke tensor A are all constant. Moreover, for any neighborhood U of p∈M, the orthornormal basis{Ei} for TM which simultaneously diagonalizes B and A at p can smoothly extend to U such that B and A are simul-taneously diagonalized on U, and the eigenvalue functionsμi of B andλi of A are all differentiable functions on U.In the fourth section, we gave the proof of our classification theorem.Main Theorem Let x:M→S(n+1)(n≥3) be an n-dimensional umbilic-free hypersurface. If x is a constant sectional hypersurface with respect to the Mobius metric g and not a generic, then the Mobius form(?)= 0, and x(M) is a Mobius isoparametric hypersurface with at most three distinct principal curvatures. Moreover, x(M) is locally Mobius equivalent to either a pre-image of a stereographic projection of the standard cylinder S(1)×R(n-1) or a pre-image of a stereographic projection of the cone spanned by the Clifford tori in S3 and o∈R4. The last case only occurs when n= 3.