• Title/Summary/Keyword: metric connection with torsion

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A NOTE ON GENERALIZED DIRAC EIGENVALUES FOR SPLIT HOLONOMY AND TORSION

  • Agricola, Ilka;Kim, Hwajeong
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.6
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    • pp.1579-1589
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    • 2014
  • We study the Dirac spectrum on compact Riemannian spin manifolds M equipped with a metric connection ${\nabla}$ with skew torsion $T{\in}{\Lambda}^3M$ in the situation where the tangent bundle splits under the holonomy of ${\nabla}$ and the torsion of ${\nabla}$ is of 'split' type. We prove an optimal lower bound for the first eigenvalue of the Dirac operator with torsion that generalizes Friedrich's classical Riemannian estimate.

A NOTE ON STATISTICAL MANIFOLDS WITH TORSION

  • Hwajeong Kim
    • Communications of the Korean Mathematical Society
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    • v.38 no.2
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    • pp.621-628
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    • 2023
  • Given a linear connection ∇ and its dual connection ∇*, we discuss the situation where ∇ + ∇* = 0. We also discuss statistical manifolds with torsion and give new examples of some type for linear connections inducing the statistical manifolds with non-zero torsion.

YANG-MILLS CONNECTIONS ON CLOSED LIE GROUPS

  • Pyo, Yong-Soo;Shin, Young-Lim;Park, Joon-Sik
    • Honam Mathematical Journal
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    • v.32 no.4
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    • pp.651-661
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    • 2010
  • In this paper, we obtain a necessary and sufficient condition for a left invariant connection in the tangent bundle over a closed Lie group with a left invariant metric to be a Yang-Mills connection. Moreover, we have a necessary and sufficient condition for a left invariant connection with a torsion-free Weyl structure in the tangent bundle over SU(2) with a left invariant Riemannian metric g to be a Yang-Mills connection.

CURVATURE OF MULTIPLY WARPED PRODUCTS WITH AN AFFINE CONNECTION

  • Wang, Yong
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.5
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    • pp.1567-1586
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    • 2013
  • In this paper, we study the Einstein multiply warped products with a semi-symmetric non-metric connection and the multiply warped products with a semi-symmetric non-metric connection with constant scalar curvature, we apply our results to generalized Robertson-Walker spacetimes with a semi-symmetric non-metric connection and generalized Kasner spacetimes with a semi-symmetric non-metric connection and find some new examples of Einstein affine manifolds and affine manifolds with constant scalar curvature. We also consider the multiply warped products with an affine connection with a zero torsion.

ON A FINSLER SPACE WITH (α, β)-METRIC AND CERTAIN METRICAL NON-LINEAR CONNECTION

  • PARK HONG-SUH;PARK HA-YONG;KIM BYUNG-DOO
    • Communications of the Korean Mathematical Society
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    • v.21 no.1
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    • pp.177-183
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    • 2006
  • The purpose of this paper is to introduce an L-metrical non-linear connection $N_j^{*i}$ and investigate a conformal change in the Finsler space with $({\alpha},\;{\beta})-metric$. The (v)h-torsion and (v)hvtorsion in the Finsler space with L-metrical connection $F{\Gamma}^*$ are obtained. The conformal invariant connection and conformal invariant curvature are found in the above Finsler space.

ON GENERALIZED FINSLER STRUCTURES WITH A VANISHING hυ-TORSION

  • Ichijyo, Yoshihiro;Lee, Il-Yong;Park, Hong-Suh
    • Journal of the Korean Mathematical Society
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    • v.41 no.2
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    • pp.369-378
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    • 2004
  • A canonical Finsler connection Nr is defined by a generalized Finsler structure called a (G, N)-structure, where G is a generalized Finsler metric and N is a nonlinear connection given in a differentiable manifold, respectively. If NT is linear, then the(G, N)-structure has a linearity in a sense and is called Berwaldian. In the present paper, we discuss what it means that NT is with a vanishing hv-torsion: ${P^{i}}\;_{jk}\;=\;0$ and introduce the notion of a stronger type for linearity of a (G, N)-structure. For important examples, we finally investigate the cases of a Finsler manifold and a Rizza manifold.

SASAKIAN STATISTICAL MANIFOLDS WITH QSM-CONNECTION AND THEIR SUBMANIFOLDS

  • Sema Kazan
    • Honam Mathematical Journal
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    • v.45 no.3
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    • pp.471-490
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    • 2023
  • In this present paper, we study QSM-connection (quarter-symmetric metric connection) on Sasakian statistical manifolds. Firstly, we express the relation between the QSM-connection ${\tilde{\nabla}}$ and the torsion-free connection ∇ and obtain the relation between the curvature tensors ${\tilde{R}}$ of ${\tilde{\nabla}}$ and R of ∇. After then we obtain these relations for ${\tilde{\nabla}}$ and the dual connection ∇* of ∇. Also, we give the relations between the curvature tensor ${\tilde{R}}$ of QSM-connection ${\tilde{\nabla}}$ and the curvature tensors R and R* of the connections ∇ and ∇* on Sasakian statistical manifolds. We obtain the relations between the Ricci tensor of QSM-connection ${\tilde{\nabla}}$ and the Ricci tensors of the connections ∇ and ∇*. After these, we construct an example of a 3-dimensional Sasakian manifold admitting the QSM-connection in order to verify our results. Finally, we study the submanifolds with the induced connection with respect to QSM-connection of statistical manifolds.

PSEUDO-HERMITIAN MAGNETIC CURVES IN NORMAL ALMOST CONTACT METRIC 3-MANIFOLDS

  • Lee, Ji-Eun
    • Communications of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.1269-1281
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    • 2020
  • In this article, we show that a pseudo-Hermitian magnetic curve in a normal almost contact metric 3-manifold equipped with the canonical affine connection ${\hat{\nabla}}^t$ is a slant helix with pseudo-Hermitian curvature ${\hat{\kappa}}={\mid}q{\mid}\;sin\;{\theta}$ and pseudo-Hermitian torsion ${\hat{\tau}}=q\;cos\;{\theta}$. Moreover, we prove that every pseudo-Hermitian magnetic curve in normal almost contact metric 3-manifolds except quasi-Sasakian 3-manifolds is a slant helix as a Riemannian geometric sense. On the other hand we will show that a pseudo-Hermitian magnetic curve γ in a quasi-Sasakian 3-manifold M is a slant curve with curvature κ = |(t - α) cos θ + q| sin θ and torsion τ = α + {(t - α) cos θ + q} cos θ. These curves are not helices, in general. Note that if the ambient space M is an α-Sasakian 3-manifold, then γ is a slant helix.