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Extra resources for Complex Analytic Differential Geometry
Alexander generalized the Hadamard theorem for compact hypersurfaces in any complete, simply connected Riemannian manifold of nonpositive sectional curvature . A topological immersion f : N n → M of a manifold N n into a Riemannian manifold M is called locally convex at a point x ∈ N n if x has a neighborhood U such that f (U ) is a part of the boundary of a convex set in M. Heijenoort proved the following theorem. Let f : N n → E n+1 , where n ≥ 2, be a topological immersion of a connected manifold N n .
From the part I) of theorem 1 it follows that F n is a compact convex hypersurface diffeomorphic to S n . 1  we obtain that every tangent sphere of radius r0 is globally supporting and F n belongs to closed balls bounded by these spheres. Two cases are possible: I) There exist two different points P1 , P2 ∈ F n such that tangent spheres S1 (r0 ), S2 (r0 ) at these points of radius r0 don’t coincide. Than F n lies at the intersection of the balls bounded of these spheres. In the Hadamard manifold the intersection of different balls of radius r0 belongs to the ball of lower radius r0 .
K2 (14) The expression on the right side is a quadratic expression with respect to |(hi bi )|. The discriminant of this polynomial is 1 | grad h|2 |b|2 − h2 sin2 ϕ|b|2 . k22 But cos2 ϕ = sin2 ϕ = k22 h2 1 = , 1 + | grad ρ|2 k22 h2 + | grad h|2 | grad h|2 . + | grad h|2 k22 h2 (15) 34 A. A. Borisenko We rewrite (15) in the form | grad h|4 + | grad h|2 |b|2 k22 k22 h2 ≥ 0. (16) From (14) it follows hi,j bi bj ≥ 0. (17) Let L be the lines on F n which satisfy the system of equations ∂ 2θ k + ∂s 2 ∂θ s = 0.