By Luther Pfahler Eisenhart
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Extra info for An introduction to differential geometry with use of the tensor calculus
Currier, On hypersurfaces of hyperbolic space inﬁnitesimally supported by horospheres, Trans. Amer. Math. , (1989), 313, (1), 412–431.  A. A. Borisenko, D. I. Vlasenko, Convex surfaces in Lobachevsky space, Math. Physics, Analysis, Geometry, (1997), 4, (3), 278–285.  S. A. Scherbakov, Regularity of radial ﬁeld on Hadamard manifold which clamped sectional curvature and bounded geometry, VINITI, dep. (3815–81), (1981), 30.  A. A. Borisenko, Convex sets in Hadamard manifolds, Differential Geometry and its Applications, (2002), 17, 111–121.
From the conditions of the theorem it follows that the normal curvatures kn /H n of any horosphere H n in M n+1 satisfy kn /H n ≤ k2 . And for every point P ∈ F n , the normal curvatures of F n and the tangent horosphere H n in the corresponding directions a satisfy the inequality kn (a)/F n ≥ kn (a)/H n . I) Suppose that at one point P0 the following strong inequality is true kn (a)/F n > kn (a)/H n . (1) Let n0 be the unit normal at the point P0 , such that the normal curvatures of F n at the point P0 ∈ F n with respect to normal n0 are positive, and H n be the 30 A.
Calvaruso Making use of formulas (14)–(18), it is possible to prove the following Proposition 1 ([CP2]). A three-dimensional semi-symmetric contact metric manifold (M 3 , η, g) satisfying A = 0 or B = 0, either is ﬂat or has constant curvature 1. We are now ready to prove the three-dimensional version of Theorem 16: Theorem 18. Let (M 2 , g) be a Riemannian surface. (T1 M 2 , g) ¯ (equivalently, (T1 M 2 , 2 gS )) is semi-symmetric if and only if (M , g) is either ﬂat or locally isometric to S 2 (1).
An introduction to differential geometry with use of the tensor calculus by Luther Pfahler Eisenhart