• 제목/요약/키워드: Levi-Civita connection

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A FAMILY OF CHARACTERISTIC CONNECTIONS

  • Kim, Hwajeong
    • 충청수학회지
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    • 제26권4호
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    • pp.843-852
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    • 2013
  • The characteristic connection is a good substitute for Levi-Civita connection in studying non-integrable geometries. In this paper we consider the homogeneous space $U(3)/(U(1){\times}U(1){\times}U(1))$ with a one-parameter family of Hermitian structures. We prove that the one-parameter family of Hermtian structures admit a characteristic connection. We also compute the torsion of the characteristic connecitons.

NORMAL HOLONOMY GROUP OF A RIEMANNIAN FOLIATIO $N^*$

  • Pak, Hong-Kyung;Pak, Jin-Suk
    • 대한수학회보
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    • 제30권1호
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    • pp.17-23
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    • 1993
  • In this paper, we will discuss on the above problem for the case that .upsilon. is a Riemannian foliation. If .upsilon. is a Riemannian foliation on (M, g), we derive some basic relations between the curvature $R^{D}$ of the normal connection D and the curvature R of the Levi-Civita connection .del. on (M, g) (see Lemma 1).).

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CURVATURES OF SEMI-SYMMETRIC METRIC CONNECTIONS ON STATISTICAL MANIFOLDS

  • Balgeshir, Mohammad Bagher Kazemi;Salahvarzi, Shiva
    • 대한수학회논문집
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    • 제36권1호
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    • pp.149-164
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    • 2021
  • By using a statistical connection, we define a semi-symmetric metric connection on statistical manifolds and study the geometry of these manifolds and their submanifolds. We show the symmetry properties of the curvature tensor with respect to the semi-symmetric metric connections. Also, we prove the induced connection on a submanifold with respect to a semi-symmetric metric connection is a semi-symmetric metric connection and the second fundamental form coincides with the second fundamental form of the Levi-Civita connection. Furthermore, we obtain the Gauss, Codazzi and Ricci equations with respect to the new connection. Finally, we construct non-trivial examples of statistical manifolds admitting a semi-symmetric metric connection.

Curvature homogeneity for four-dimensional manifolds

  • Sekigawa, Kouei;Suga, Hiroshi;Vanhecke, Lieven
    • 대한수학회지
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    • 제32권1호
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    • pp.93-101
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    • 1995
  • Let (M,g) be an n-dimensional, connected Riemannian manifold with Levi Civita connection $\nabla$ and Riemannian curvature tensor R defined by $$ R_XY = [\nabla_X, \nabla_Y] - \nabla_{[X,Y]} $$ for all smooth vector fields X, Y. $\nablaR, \cdots, \nabla^kR, \cdots$ denote the successive covariant derivatives and we assume $\nabla^0R = R$.

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ALMOST EINSTEIN MANIFOLDS WITH CIRCULANT STRUCTURES

  • Dokuzova, Iva
    • 대한수학회지
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    • 제54권5호
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    • pp.1441-1456
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    • 2017
  • We consider a 3-dimensional Riemannian manifold M with a circulant metric g and a circulant structure q satisfying $q^3=id$. The structure q is compatible with g such that an isometry is induced in any tangent space of M. We introduce three classes of such manifolds. Two of them are determined by special properties of the curvature tensor. The third class is composed by manifolds whose structure q is parallel with respect to the Levi-Civita connection of g. We obtain some curvature properties of these manifolds (M, g, q) and give some explicit examples of such manifolds.

BERGER TYPE DEFORMED SASAKI METRIC ON THE COTANGENT BUNDLE

  • Zagane, Abderrahim
    • 대한수학회논문집
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    • 제36권3호
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    • pp.575-592
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    • 2021
  • In this paper, we introduce the Berger type deformed Sasaki metric on the cotangent bundle T*M over an anti-paraKähler manifold (M, 𝜑, g) as a new natural metric with respect to g non-rigid on T*M. Firstly, we investigate the Levi-Civita connection of this metric. Secondly, we study the curvature tensor and also we characterize the scalar curvature.

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

  • Djaa, Nour Elhouda;Zagane, Abderrahim
    • 대한수학회논문집
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    • 제37권3호
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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.

Notes on the Second Tangent Bundle over an Anti-biparaKaehlerian Manifold

  • Nour Elhouda Djaa;Aydin Gezer
    • Kyungpook Mathematical Journal
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    • 제63권1호
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    • pp.79-95
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    • 2023
  • In this note, we define a Berger type deformed Sasaki metric as a natural metric on the second tangent bundle of a manifold by means of a biparacomplex structure. First, we obtain the Levi-Civita connection of this metric. Secondly, we get the curvature tensor, sectional curvature, and scalar curvature. Afterwards, we obtain some formulas characterizing the geodesics with respect to the metric on the second tangent bundle. Finally, we present the harmonicity conditions for some maps.

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

  • Abderrahim Zagane
    • 대한수학회논문집
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    • 제38권4호
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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.