• Title, Summary, Keyword: totally geodesic

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A note on totally geodesic maps

  • Chung, In-Jae;Koh, Sung-Eun
    • Bulletin of the Korean Mathematical Society
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    • v.29 no.2
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    • pp.233-236
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    • 1992
  • Let f:M.rarw.N be a smooth map between Rioemannian manifolds M and N. If f maps geodesics of M to geodesics of N, f is called totally geodesic. As is well known, totally geodesic maps are harmonic and the image f(M) of a totally geodesic map f:M.rarw. N is an immersed totally geodesic submanifold of N (cf. .cint. 6.3 of [W]). We are interested in the following question: When is a harmonic map f:M .rarw. N with rank .leq. 1 everywhere on M totally geodesic\ulcorner In other words, when is the image of a harmonic map f:M .rarw. N with rank .leq. 1 everywhere on M geodesics of N\ulcorner In this note, we give some sufficient conditions on curvatures of M. It is interesting that no curvature assumptions on target manifolds are necessary in Theorems 1 and 2. Some properties of totally geodesic maps are also given in Theorem 3. We think our Theorem 3 is somewhat unusual in view of the following classical theorem of Eells and Sampson (see pp.124 of [ES]).

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HORIZONTALLY HOMOTHETIC HARMONIC MORPHISMS AND STABILITY OF TOTALLY GEODESIC SUBMANIFOLDS

  • Yun, Gab-Jin;Choi, Gun-Don
    • Journal of the Korean Mathematical Society
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    • v.45 no.2
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    • pp.493-511
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    • 2008
  • In this article, we study the relations of horizontally homothetic harmonic morphisms with the stability of totally geodesic submanifolds. Let $\varphi:(M^n,g)\rightarrow(N^m,h)$ be a horizontally homothetic harmonic morphism from a Riemannian manifold into a Riemannian manifold of non-positive sectional curvature and let T be the tensor measuring minimality or totally geodesics of fibers of $\varphi$. We prove that if T is parallel and the horizontal distribution is integrable, then for any totally geodesic submanifold P in N, the inverse set, $\varphi^{-1}$(P), is volume-stable in M. In case that P is a totally geodesic hypersurface the condition on the curvature can be weakened to Ricci curvature.

NON-EXISTENCE OF TOTALLY GEODESIC SCREEN DISTRIBUTIONS ON LIGHTLIKE HYPERSURFACES OF INDEFINITE KENMOTSU MANIFOLDS

  • Jin, Dae Ho
    • Communications of the Korean Mathematical Society
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    • v.28 no.2
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    • pp.353-360
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    • 2013
  • We study lightlike hypersurfaces of indefinite Kenmotsu manifolds. The purpose of this paper is to prove that there do not exist totally geodesic screen distributions on semi-symmetric lightlike hypersurfaces of indefinite Kenmotsu manifolds with flat transversal connection.

Totally Umbilical Slant Lightlike Submanifolds of Indefinite Kaehler Manifolds

  • Sachdeva, Rashmi;Kumar, Rakesh;Bhatia, Satvinder Singh
    • Kyungpook Mathematical Journal
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    • v.57 no.3
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    • pp.503-516
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    • 2017
  • In this paper, we study totally umbilical slant lightlike submanifolds of indefinite Kaehler manifolds. We prove that there do not exist totally umbilical proper slant lightlike submanifolds in indefinite Kaehler manifolds other than totally geodesic proper slant lightlike submanifolds. We also prove that there do not exist totally umbilical proper slant lightlike submanifolds of indefinite Kaehler space forms. Finally, we give a characterization theorem on minimal slant lightlike submanifolds.

KÄHLER SUBMANIFOLDS WITH LOWER BOUNDED TOTALLY REAL BISECTIONL CURVATURE TENSOR II

  • Pyo, Yong-Soo;Shin, Kyoung-Hwa
    • Communications of the Korean Mathematical Society
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    • v.17 no.2
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    • pp.279-293
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    • 2002
  • In this paper, we prove that if every totally real bisectional curvature of an n($\geq$3)-dimensional complete Kahler submanifold of a complex projective space of constant holomorphic sectional curvature c is greater than (equation omitted) (3n$^2$+2n-2), then it is totally geodesic and compact.

LIGHTLIKE HYPERSURFACES OF A SEMI-RIEMANNIAN MANIFOLD OF QUASI-CONSTANT CURVATURE

  • Jin, Dae-Ho
    • Communications of the Korean Mathematical Society
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    • v.27 no.4
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    • pp.763-770
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    • 2012
  • In this paper, we study the geometry lightlike hypersurfaces (M, $g$, S(TM)) of a semi-Riemannian manifold ($\tilde{M}$, $\tilde{g}$) of quasi-constant curvature subject to the conditions: (1) The curvature vector field of $\tilde{M}$ is tangent to M, and (2) the screen distribution S(TM) is either totally geodesic in M or totally umbilical in $\tilde{M}$.

SEMI-INVARIANT SUBMANIFOLDS OF (LCS)n-MANIFOLD

  • Bagewadi, Channabasappa Shanthappa;Nirmala, Dharmanaik;Siddesha, Mallannara Siddalingappa
    • Communications of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.1331-1339
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    • 2018
  • In this paper the decomposition of basic equations of $(LCS)_n$-manifolds is carried out into horizontal and vertical projections. Further, we study the integrability of the distributions $D,D{\oplus}[{\xi}]$ and $D^{\perp}$ totally geodesic of semi-invariant submanifolds of $(LCS)_n$-manifold.

CERTAIN RESULTS ON SUBMANIFOLDS OF GENERALIZED SASAKIAN SPACE-FORMS

  • Yadav, Sunil Kumar;Chaubey, Sudhakar K
    • Honam Mathematical Journal
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    • v.42 no.1
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    • pp.123-137
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    • 2020
  • The object of the present paper is to study certain geometrical properties of the submanifolds of generalized Sasakian space-forms. We deduce some results related to the invariant and anti-invariant slant submanifolds of the generalized Sasakian spaceforms. Finally, we study the properties of the sectional curvature, totally geodesic and umbilical submanifolds of the generalized Sasakian space-forms. To prove the existence of almost semiinvariant and anti-invariant submanifolds, we provide the non-trivial examples.

THE FUNDAMENTAL FORMULAS OF FINSLER SUBMANIFOLDS

  • Li, Jintang
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.4
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    • pp.767-775
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    • 2010
  • Let ${\varphi}\;:\;(M^n,\;F)\;{\rightarrow}\;(\overline{M}^{n+p},\;\overline{F})$ be an isometric immersion from a Finsler manifold to a Finsler manifold. In this paper, we shall obtain the Gauss and Codazzi equations with respect to the Chern connection on submanifolds M, by which we prove that if M is a weakly totally geodesic submanifold of $\overline{M}$, then flag curvature of M equals flag curvature of $\overline{M}$.