• 제목/요약/키워드: the covariant derivative

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크리스토펠, 리치, 레비-치비타에 의한 19세기 중반부터 20세기 초반까지 미분기하학의 발전 (On the Development of Differential Geometry from mid 19C to early 20C by Christoffel, Ricci and Levi-Civita)

  • 원대연
    • 한국수학사학회지
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    • 제28권2호
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    • pp.103-115
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    • 2015
  • Contemporary differential geometry owes much to the theory of connections on the bundles over manifolds. In this paper, following the work of Gauss on surfaces in 3 dimensional space and the work of Riemann on the curvature tensors on general n dimensional Riemannian manifolds, we will investigate how differential geometry had been developed from mid 19th century to early 20th century through lives and mathematical works of Christoffel, Ricci-Curbastro and Levi-Civita. Christoffel coined the Christoffel symbol and Ricci used the Christoffel symbol to define the notion of covariant derivative. Levi-Civita completed the theory of absolute differential calculus with Ricci and discovered geometric meaning of covariant derivative as parallel transport.

A NEW CHARACTERIZATION OF RULED REAL HYPERSURFACES IN COMPLEX SPACE FORMS

  • Ahn, Seong-Soo;Choi, Young-Suk;Suh, Young-Jin
    • 대한수학회보
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    • 제36권3호
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    • pp.513-532
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    • 1999
  • The purpose of this paper is to give another new characterization of ruled real hypersurfaces in a complex space form $M_n$(c), c$\neq$0 in terms of the covariant derivative of its Weingarten map in the direction of the structure vector $\xi$.

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Real Hypersurfaces with k-th Generalized Tanaka-Webster Connection in Complex Grassmannians of Rank Two

  • Jeong, Imsoon;Lee, Hyunjin
    • Kyungpook Mathematical Journal
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    • 제57권3호
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    • pp.525-535
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    • 2017
  • In this paper, we consider two kinds of derivatives for the shape operator of a real hypersurface in a $K{\ddot{a}}hler$ manifold which are named the Lie derivative and the covariant derivative with respect to the k-th generalized Tanaka-Webster connection ${\hat{\nabla}}^{(k)}$. The purpose of this paper is to study Hopf hypersurfaces in complex Grassmannians of rank two, whose Lie derivative of the shape operator coincides with the covariant derivative of it with respect to ${\hat{\nabla}}^{(k)}$ either in direction of any vector field or in direction of Reeb vector field.

ON THE LIE DERIVATIVE OF REAL HYPERSURFACES IN ℂP2 AND ℂH2 WITH RESPECT TO THE GENERALIZED TANAKA-WEBSTER CONNECTION

  • PANAGIOTIDOU, KONSTANTINA;PEREZ, JUAN DE DIOS
    • 대한수학회보
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    • 제52권5호
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    • pp.1621-1630
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    • 2015
  • In this paper the notion of Lie derivative of a tensor field T of type (1,1) of real hypersurfaces in complex space forms with respect to the generalized Tanaka-Webster connection is introduced and is called generalized Tanaka-Webster Lie derivative. Furthermore, three dimensional real hypersurfaces in non-flat complex space forms whose generalized Tanaka-Webster Lie derivative of 1) shape operator, 2) structure Jacobi operator coincides with the covariant derivative of them with respect to any vector field X orthogonal to ${\xi}$ are studied.

ON THE FINSLER SPACES WITH f-STRUCTURE

  • Park, Hong-Suh;Lee, Il-Yong
    • 대한수학회보
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    • 제36권2호
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    • pp.217-224
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    • 1999
  • In this paper the properties of the Finsler metrics compatible with an f-structure are investigated.

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ON SOME SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 IN A COMPLEX PROJECTIVE SPACE

  • Lee, Seong-Baek;Kim, Soo-Jin
    • 대한수학회논문집
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    • 제18권2호
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    • pp.309-323
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    • 2003
  • In this paper, We characterize a semi-invariant sub-manifold of codimension 3 satisfying ∇$\varepsilon$A = 0 in a complex projective space CP$\^$n+1/, where ∇$\varepsilon$A is the covariant derivative of the shape operator A in the direction of the distinguished normal with respect to the structure vector field $\varepsilon$.

ON $\eta$K-CONFORMAL KILLING TENSOR IN COSYMPLECTIC MANIFOLD WITH VANISHING COSYMPLECTIC BOCHNER CURVATURE TENSOR$^*$

  • Jun, Jae-Bok;Kim, Un-Kyu
    • 대한수학회보
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    • 제32권1호
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    • pp.25-34
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    • 1995
  • S. Tachibana [10] has defined a confornal Killing tensor in a n-dimensional Riemannian manifold M by a skew symmetric tensor $u_[ji}$ satisfying the equation $$ \nabla_k u_{ji} + \nabla_j u_{ki} = 2\rho_i g_{kj} - \rho_j g_{ki} - \rho_k g_{ji}, $$ where $g_{ji}$ is the metric tensor of M, $\nabla$ denotes the covariant derivative with respect to $g_{ji}$ and $\rho_i$ is a associated covector field of $u_{ji}$. In here, a covector field means a 1-form.

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On generic submanifolds of a complex projective space

  • Seong Baek Lee;Seung Gook Han;Nam Gil Kim;Seong Soo Ahn
    • 대한수학회논문집
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    • 제11권3호
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    • pp.743-756
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    • 1996
  • The purpose of this paper is to compute the covariant derivative of a shape operator of a generic submanifold of a complex space form without using the Green-Stoke's theorem. In particular, we classify complete generic submanifolds of a complex number space $C^m$ with parallel mean curvature vector satisfying a certain condition.

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LINEAR CONNECTIONS IN THE BUNDLE OF LINEAR FRAMES

  • Park, Joon-Sik
    • 충청수학회지
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    • 제25권4호
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    • pp.731-738
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    • 2012
  • Let L(M) be the bundle of all linear frames over $M,\;u$ an arbitrarily given point of L(M), and ${\nabla}\;:\;\mathfrak{X}(M)\;{\times}\;\mathfrak{X}(M)\;\rightarrow\;\mathfrak{X}(M)$ a linear connection on L(M). Then the following results are well known: the horizontal subspace and the connection form at the point $u$ may be written in terms of local coordinates of $u\;{\epsilon}\;L(M)$ and Christoffel's symbols defined by $\nabla$. These results are very fundamental on the study of the theory of connections. In this paper we show that the local expressions of those at the point $u$ do not depend on the choice of a local coordinate system around the point $u\;{\epsilon}\;L(M)$, which is rarely seen. Moreover we give full explanations for the following fact: the covariant derivative on M which is defined by the parallelism on L(M), determined from the connection form above, coincides with $\nabla$.