# An introduction to differential geometry with use of the by Luther Pfahler Eisenhart PDF

By Luther Pfahler Eisenhart

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Currier, On hypersurfaces of hyperbolic space inﬁnitesimally supported by horospheres, Trans. Amer. Math. , (1989), 313, (1), 412–431. [6] A. A. Borisenko, D. I. Vlasenko, Convex surfaces in Lobachevsky space, Math. Physics, Analysis, Geometry, (1997), 4, (3), 278–285. [7] S. A. Scherbakov, Regularity of radial ﬁeld on Hadamard manifold which clamped sectional curvature and bounded geometry, VINITI, dep. (3815–81), (1981), 30. [8] 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

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