• Title/Summary/Keyword: scalar curvature

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THE EXPANSION OF MEAN DISTANCE OF BROWNIAN MOTION ON RIEMANNIAN MANIFOLD

  • Kim, Yoon-Tae;Park, Hyun-Suk;Jeon, Jong-Woo
    • Proceedings of the Korean Statistical Society Conference
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    • 2003.05a
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    • pp.37-42
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    • 2003
  • We study the asymptotic expansion in small time of the mean distance of Brownian motion on Riemannian manifolds. We compute the first four terms of the asymptotic expansion of the mean distance by using the decomposition of Laplacian into homogeneous components. This expansion can he expressed in terms of the scalar valued curvature invariants of order 2, 4, 6.

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SOME RESULTS ON THE GEOMETRY OF A NON-CONFORMAL DEFORMATION OF A METRIC

  • Djaa, Nour Elhouda;Zagane, Abderrahim
    • Communications of the Korean Mathematical Society
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    • v.37 no.3
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    • pp.865-879
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    • 2022
  • Let (Mm, g) be an m-dimensional Riemannian manifold. In this paper, we introduce a new class of metric on (Mm, g), obtained by a non-conformal deformation of the metric g. First we investigate the Levi-Civita connection of this metric. Secondly we characterize the Riemannian curvature, the sectional curvature and the scalar curvature. In the last section we characterizes some class of proper biharmonic maps. Examples of proper biharmonic maps are constructed when (Mm, g) is an Euclidean space.

GEOMETRY OF GENERALIZED BERGER-TYPE DEFORMED METRIC ON B-MANIFOLD

  • Abderrahim Zagane
    • Communications of the Korean Mathematical Society
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    • v.38 no.4
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    • pp.1281-1298
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    • 2023
  • Let (M2m, 𝜑, g) be a B-manifold. In this paper, we introduce a new class of metric on (M2m, 𝜑, g), obtained by a non-conformal deformation of the metric g, called a generalized Berger-type deformed metric. First we investigate the Levi-Civita connection of this metric. Secondly we characterize the Riemannian curvature, the sectional curvature and the scalar curvature. Finally, we study the proper biharmonicity of the identity map and of a curve on M with respect to a generalized Berger-type deformed metric.

SCALAR CURVATURE DECREASE FROM A HYPERBOLIC METRIC

  • Kang, Yutae;Kim, Jongsu
    • The Pure and Applied Mathematics
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    • v.20 no.4
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    • pp.269-276
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    • 2013
  • We find an explicit $C^{\infty}$-continuous path of Riemannian metrics $g_t$ on the 4-d hyperbolic space $\mathbb{H}^4$, for $0{\leq}t{\leq}{\varepsilon}$ for some number ${\varepsilon}$ > 0 with the following property: $g_0$ is the hyperbolic metric on $\mathbb{H}^4$, the scalar curvatures of $g_t$ are strictly decreasing in t in an open ball and $g_t$ is isometric to the hyperbolic metric in the complement of the ball.

α-SCALAR CURVATURE OF THE t-MANIFOLD

  • Cho, Bong-Sik;Jung, Sun-Young
    • Honam Mathematical Journal
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    • v.33 no.4
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    • pp.487-493
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    • 2011
  • The Fisher information matrix plays a significant role in statistical inference in connection with estimation and properties of variance of estimators. In this paper, we define the parameter space of the t-manifold using its Fisher's matrix and characterize the t-manifold from the viewpoint of information geometry. The ${\alpha}$-scalar curvatures to the t-manifold are calculated.

A THEOREM OF G-INVARIANT MINIMAL HYPERSURFACES WITH CONSTANT SCALAR CURVATURES IN Sn+1

  • So, Jae-Up
    • Honam Mathematical Journal
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    • v.31 no.3
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    • pp.381-398
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    • 2009
  • Let $G\;=\;O(k){\times}O(k){\times}O(q)$ and let $M^n$ be a closed G-invariant minimal hypersurface with constant scalar curvature in $S^{n+1}$. Then we obtain a theorem: If $M^n$ has 2 distinct principal curvatures at some point p, then the square norm of the second fundamental form of $M^n$, S = n.

PARTIAL DIFFERENTIAL EQUATIONS AND SCALAR CURVATURE ON SEMIRIEMANNIAN MANIFOLDS(I)

  • Jung, Yoon-Tae;Kim, Yun-Jeong;Lee, Soo-Young;Shin, Cheol-Guen
    • The Pure and Applied Mathematics
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    • v.5 no.2
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    • pp.115-122
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    • 1998
  • In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct timelike or null future(or past) complete Lorentzian metrics on $M{\;}={\;}[a,{\;}{\infty}){\times}_f{\;}N$ with specific scalar curvatures.

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