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Let M ? S~4 be a complete orientable hypersurface with constant scalar curvature. For any v ∈ R~5, let us define the two real functions l_v,f_v:M →R on M by l_v(x) = and fv(x) = |ν(x), v- with ν:M →S4 a Gauss map of M. In th...
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Let M ? S~4 be a complete orientable hypersurface with constant scalar curvature. For any v ∈ R~5, let us define the two real functions l_v,f_v:M →R on M by l_v(x) = and fv(x) = |ν(x), v- with ν:M →S4 a Gauss map of M. In this paper, we show that if we have that l_v = λf_v for some nonzero vector v ∈ R~5 and some real number λ, then M is either totally umbilical (a Euclidean sphere) or M is a Cartesian product of Euclidean spheres. We will also show with an example that the completeness condition is needed in this theorem.
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The Riemannian manifolds equipped with semi-symmetric metric connection have been studied by O.C. Andonie (1976), M.C. Chaki and A. Konar (1981), U.C. De (1991), U.C. De and S.C. Biswas (1997) and others. P.N. Pandey and S.K. Dube...
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The Riemannian manifolds equipped with semi-symmetric metric connection have been studied by O.C. Andonie (1976), M.C. Chaki and A. Konar (1981), U.C. De (1991), U.C. De and S.C. Biswas (1997) and others. P.N. Pandey and S.K. Dubey (2004) discussed an almost Grayan manifold equipped with semi-symmetric metric connection while a Kahler manifold equipped with semi-symmetric metric connection has been studied by P.N. Pandey and B.B. Chaturvedi (2007). An algebraic characterization of an almost Hermitian manifold of pointwise constant anti-holomorphic sectional curvature has been obtained by G. Ganchev (1987). A 4-dimensional almost Hermitian manifold of pointwise constant anti-holomorphic sectional curvature was studied by Takuji Sato (2003). Takuji Sato obtained the necessary and sufficient condition for a 4-dimensional almost Hermitian manifold to have pointwise constant anti-holomorphic sectional curvature. The aim of the present paper is to discuss the relation between curvature tensors with respect to a semi-symmetric metric connection and a Riemannian connection in a '4-dimensional almost Hermite manifold. The relation between complexification of curvature tensors with respect to a semi-symmetric metric connection and a Riemannian connection has also been obtained. We have also discussed the self duality and conformally flat case in a 4-dimensional almost Hermitian manifold equipped with semi-symmetric metric connection.
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The Ribaucour transformation is applied to the family of Lagrangian submanifolds of dimension n and nonzero constant sectional curvature c of complex space forms of complex dimension n and constant holomorphic sectional curvature ...
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The Ribaucour transformation is applied to the family of Lagrangian submanifolds of dimension n and nonzero constant sectional curvature c of complex space forms of complex dimension n and constant holomorphic sectional curvature 4c. As a consequence, a process is obtained to generate a new family of such submanifolds starting from a given one. In particular, explicit parameterizations in terms of elementary functions of examples with arbitrary dimension and curvature are provided. A permutability formula is derived which provides a simple algebraic procedure to construct further examples once two Ribaucour transforms of a given submanifold are known. The analytical counterparts of the above results are also discussed.
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This paper studies Lp - version of the Hardy type inequalities on the geodesic sphereof constant sectional curvature and establishes that the corresponding constant is sharp. Furthermore, the inequalities obtained are used to deri...
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This paper studies Lp - version of the Hardy type inequalities on the geodesic sphereof constant sectional curvature and establishes that the corresponding constant is sharp. Furthermore, the inequalities obtained are used to derive an uncertainty principle inequality and anotherinequality involving the first nonzero eigenvalue of the p-Laplacian on the sphere.
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It is shown that a connected Riemannian manifold has constant sectional curvature if and only if every one of its points is a non-degenerate maximum of some germ of smooth functions whose Riemannian gradient is a conformal vector field.
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We study affine hyperspheres with constant sectional curvature. More precisely we obtain a classification of the affine hyperspheres with constant sectional curvature c, provided c ≠ H, where H denotes the affine mean curvature o...
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We study affine hyperspheres with constant sectional curvature. More precisely we obtain a classification of the affine hyperspheres with constant sectional curvature c, provided c ≠ H, where H denotes the affine mean curvature of the immersion. Our classification gives a complete and positive answer to a conjecture of M. Magid and P. Ryan about these hyperspheres.
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Abstract The concept of Hessian sectional curvature of a Hessian manifold M was introduced by Shima as a real-analogue of the holomorphic sectional curvature in K?hler geometry [15]. The former, unlike the latter, is also well-def...
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Abstract The concept of Hessian sectional curvature of a Hessian manifold M was introduced by Shima as a real-analogue of the holomorphic sectional curvature in K?hler geometry [15]. The former, unlike the latter, is also well-defined when the real dimension of M is 1. In this case, the Hessian sectional curvature is just a real-valued function on M. In this paper, we give a complete classification of 1-dimensional exponential families E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {E}}$$\end{document} defined over a finite set Ω={x0,…,xm}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega =\{x_{0},\ldots ,x_{m}\}$$\end{document} whose Hessian sectional curvature is constant. We observe an interesting phenomenon: if E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {E}}$$\end{document} has constant Hessian sectional curvature, say λ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ,$$\end{document} then λ=-1k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda =-\tfrac{1}{k}$$\end{document} for some integer 1≤k≤m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le k\le m$$\end{document}. We show that the family of Binomial distributions plays a central role in this classification.
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It is known that the minimal 3-spheres of CR type with constant sectional curvature have been classified explicitly, and also that the weakly Lagrangian case has been studied. In this paper, we provide some examples of minimal 3-s...
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It is known that the minimal 3-spheres of CR type with constant sectional curvature have been classified explicitly, and also that the weakly Lagrangian case has been studied. In this paper, we provide some examples of minimal 3-spheres with constant curvature in the complex projective space, which are neither of CR type nor weakly Lagrangian, and give the adapted frame of a minimal 3-sphere of CR type with constant sectional curvature.
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For a two-dimensional oriented Riemannian manifold (M, g), we consider the horizontal and vertical surfaces immersed in the 3-dimensional Riemannian manifold SO(M, g), the total space of positive orthonormal frame bundle over (M, ...
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For a two-dimensional oriented Riemannian manifold (M, g), we consider the horizontal and vertical surfaces immersed in the 3-dimensional Riemannian manifold SO(M, g), the total space of positive orthonormal frame bundle over (M, g), endowed by the Wagner lift metric (g) over cap. The horizontal surfaces are the sections of SO(M, g), and the vertical surfaces are the preimages of regular curves of M. We find the first and second fundamental form of the horizontal and vertical surfaces, and the mean curvature of these surfaces. Using these results, we exhibit suitable conditions for the existence of minimal surfaces and constant mean curvature surfaces in SO(M, g).
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