Research partially supported by CNPq, Brasil ff d = -S Cdv (Sect. 1). However, d is not necessarily an integer, but 3d is since the first Chern form ~M is 1 3~r c1= ~n/(f~11 +t222) = O"where ~ denotes the K/ihler form of P~, and so 3 -~ represents an integral cohomology class of p~.
Using the notion of the ellipse of curvature we study compact surfaces in high dimensional space forms. We obtain some inequalities relating the area of the surface and the integral of the square of the norm of the mean curvature vector with topological invariants. In certain cases, the ellipse is a circle; when this happens, restrictions on the Gaussian and normal curvatures give us some rigidity results.
In this paper, we prove that an n-dimensional closed minimal hypersurface M with Ricci curvature Ric(M) ≥ n 2 of a unit sphere S n+1 (1) is isometric to a Clifford torus if n ≤ S ≤ n + 14(n+4) 9n+30 , where S is the squared norm of the second fundamental form of M .
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