• 제목/요약/키워드: Levi-Civita connection

검색결과 31건 처리시간 0.023초

AFFINE YANG-MILLS CONNECTIONS ON NORMAL HOMOGENEOUS SPACES

  • Park, Joon-Sik
    • 호남수학학술지
    • /
    • 제33권4호
    • /
    • pp.557-573
    • /
    • 2011
  • Let G be a compact and connected semisimple Lie group, H a closed subgroup, g (resp. h) the Lie algebra of G (resp. H), B the Killing form of g, g the normal metric on the homogeneous space G/H which is induced by -B. Let D be an invarint connection with Weyl structure (D, g, ${\omega}$) in the tangent bundle over the normal homogeneous Riemannian manifold (G/H, g) which is projectively flat. Then, the affine connection D on (G/H, g) is a Yang-Mills connection if and only if D is the Levi-Civita connection on (G/H, g).

LIGHTLIKE SUBMANIFOLDS OF A SEMI-RIEMANNIAN MANIFOLD WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION

  • Yucesan, Ahmet;Yasar, Erol
    • 대한수학회보
    • /
    • 제47권5호
    • /
    • pp.1089-1103
    • /
    • 2010
  • In this paper, we study lightlike submanifolds of a semi-Riemannian manifold admitting a semi-symmetric non-metric connection. We obtain a necessary and a sufficient condition for integrability of the screen distribution. Then we give the conditions under which the Ricci tensor of a lightlike submanifold with a semi-symmetric non-metric connection is symmetric. Finally, we show that the Ricci tensor of a lightlike submanifold of semi-Riemannian space form is not parallel with respect to the semi-symmetric non-metric connection.

YAMABE SOLITONS ON KENMOTSU MANIFOLDS

  • Hui, Shyamal Kumar;Mandal, Yadab Chandra
    • 대한수학회논문집
    • /
    • 제34권1호
    • /
    • pp.321-331
    • /
    • 2019
  • The present paper deals with a study of infinitesimal CL-transformations on Kenmotsu manifolds, whose metric is Yamabe soliton and obtained sufficient conditions for such solitons to be expanding, steady and shrinking. Among others, we find a necessary and sufficient condition of a Yamabe soliton on Kenmotsu manifold with respect to CL-connection to be Yamabe soliton on Kenmotsu manifold with respect to Levi-Civita connection. We found the necessary and sufficient condition for the Yamabe soliton structure to be invariant under Schouten-Van Kampen connection. Finally, we constructed an example of steady Yamabe soliton on 3-dimensional Kenmotsu manifolds with respect to Schouten-Van Kampen connection.

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

  • Jeong, Imsoon;Lee, Hyunjin
    • Kyungpook Mathematical Journal
    • /
    • 제57권3호
    • /
    • pp.525-535
    • /
    • 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.

SECOND ORDER TANGENT VECTORS IN RIEMANNIAN GEOMETRY

  • Kwon, Soon-Hak
    • 대한수학회지
    • /
    • 제36권5호
    • /
    • pp.959-1008
    • /
    • 1999
  • This paper considers foundational issues related to connections in the tangent bundle of a manifold. The approach makes use of second order tangent vectors, i.e., vectors tangent to the tangent bundle. The resulting second order tangent bundle has certain properties, above and beyond those of a typical tangent bundle. In particular, it has a natural secondary vector bundle structure and a canonical involution that interchanges the two structures. The involution provides a nice way to understand the torsion of a connection. The latter parts of the paper deal with the Levi-Civita connection of a Riemannian manifold. The idea is to get at the connection by first finding its.spary. This is a second order vector field that encodes the second order differential equation for geodesics. The paper also develops some machinery involving lifts of vector fields form a manifold to its tangent bundle and uses a variational approach to produce the Riemannian spray.

  • PDF

ALMOST QUASI-YAMABE SOLITONS ON LORENTZIAN CONCIRCULAR STRUCTURE MANIFOLDS-[(LCS)n]

  • Jun, Jae-Bok;Siddiqi, Mohd. Danish
    • 호남수학학술지
    • /
    • 제42권3호
    • /
    • pp.521-536
    • /
    • 2020
  • The object of the present paper is to study of Almost Quasi-Yamabe solitons and gradient almost quasi-Yamabe solitons on an Lorentzian concircular structure manifolds briefly say (LCS)n-manifolds under infinitesimal CL-transformations and obtained sufficient conditions for such solitons to be expanding, steady and shrinking. Also we obtained a necessary and sufficient condition of an almost quasi-Yamabe soliton with respect to the CL-connection to be an almost quasi-Yamabe soliton on (LCS)n-manifolds with respect to Levi-Civita connection. Finally, we construct an example of steady almost quasi-Yamabe soliton on 3-dimensional (LCS)n-manifolds.

루마니아 핀슬러 기하학파 형성의 역사 (On the History of Formation of Romanian School of Finsler Geometry)

  • 원대연
    • 한국수학사학회지
    • /
    • 제32권1호
    • /
    • pp.1-15
    • /
    • 2019
  • We divide the timeline of the history of Finsler geometry, which dates back to Riemann's inaugural lecture in 1854, into three periods (hibernation, hiatus, rebirth) and we study formation of Romanian Finsler school around Iasi, Romania during the hiatus period. We look for the history centered around Radu Miron who is a third generation geometer of Iasi University and the mathematical heritage there through five generations. We also investigate mathematical impact of T. Levi-Civita, D. Hilbert, ${\acute{E}}$ Cartan who are considered as top mathematicians at their time.

STATIC AND RELATED CRITICAL SPACES WITH HARMONIC CURVATURE AND THREE RICCI EIGENVALUES

  • Kim, Jongsu
    • 대한수학회지
    • /
    • 제57권6호
    • /
    • pp.1435-1449
    • /
    • 2020
  • In this article we make a local classification of n-dimensional Riemannian manifolds (M, g) with harmonic curvature and less than four Ricci eigenvalues which admit a smooth non constant solution f to the following equation $$(1)\hspace{20}{\nabla}df=f(r-{\frac{R}{n-1}}g)+x{\cdot} r+y(R)g,$$ where ∇ is the Levi-Civita connection of g, r is the Ricci tensor of g, x is a constant and y(R) a function of the scalar curvature R. Indeed, we showed that, in a neighborhood V of each point in some open dense subset of M, either (i) or (ii) below holds; (i) (V, g, f + x) is a static space and isometric to a domain in the Riemannian product of an Einstein manifold N and a static space (W, gW, f + x), where gW is a warped product metric of an interval and an Einstein manifold. (ii) (V, g) is isometric to a domain in the warped product of an interval and an Einstein manifold. For the proof we use eigenvalue analysis based on the Codazzi tensor properties of the Ricci tensor.

ON RICCI CURVATURES OF LEFT INVARIANT METRICS ON SU(2)

  • Pyo, Yong-Soo;Kim, Hyun-Woong;Park, Joon-Sik
    • 대한수학회보
    • /
    • 제46권2호
    • /
    • pp.255-261
    • /
    • 2009
  • In this paper, we shall prove several results concerning Ricci curvature of a Riemannian manifold (M, g) := (SU(2), g) with an arbitrary given left invariant metric g. First of all, we obtain the maximum (resp. minimum) of {r(X) := Ric(X,X) | ${||X||}_g$ = 1,X ${\in}$ X(M)}, where Ric is the Ricci tensor field on (M, g), and then get a necessary and sufficient condition for the Levi-Civita connection ${\nabla}$ on the manifold (M, g) to be projectively flat. Furthermore, we obtain a necessary and sufficient condition for the Ricci curvature r(X) to be always positive (resp. negative), independently of the choice of unit vector field X.

A NEW CLASSIFICATION OF REAL HYPERSURFACES WITH REEB PARALLEL STRUCTURE JACOBI OPERATOR IN THE COMPLEX QUADRIC

  • Lee, Hyunjin;Suh, Young Jin
    • 대한수학회지
    • /
    • 제58권4호
    • /
    • pp.895-920
    • /
    • 2021
  • In this paper, first we introduce the full expression of the Riemannian curvature tensor of a real hypersurface M in the complex quadric Qm from the equation of Gauss and some important formulas for the structure Jacobi operator Rξ and its derivatives ∇Rξ under the Levi-Civita connection ∇ of M. Next we give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, ∇ξRξ = 0, in the complex quadric Qm for m ≥ 3. In addition, we also consider a new notion of 𝒞-parallel structure Jacobi operator of M and give a nonexistence theorem for Hopf real hypersurfaces with 𝒞-parallel structure Jacobi operator in Qm, for m ≥ 3.