• Title/Summary/Keyword: Riemannian Geometry

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ANTI-INVARIANT SUBMERSIONS FROM ALMOST PARACONTACT RIEMANNIAN MANIFOLDS

  • Gunduzalp, Yilmaz
    • Honam Mathematical Journal
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    • v.41 no.4
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    • pp.769-780
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    • 2019
  • We introduce anti-invariant Riemannian submersions from almost paracontact Riemannian manifolds onto Riemannian manifolds. We give an example, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion and check the harmonicity of such submersions.

GEOMETRY OF HALF LIGHTLIKE SUBMANIFOLDS OF A SEMI-RIEMANNIAN SPACE FORM WITH A SEMI-SYMMETRIC METRIC CONNECTION

  • Jin, Dae Ho
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.4
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    • pp.769-781
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    • 2011
  • We study the geometry of half lightlike sbmanifolds M of a semi-Riemannian space form $\tilde{M}(c)$ admitting a semi-symmetric metric connection subject to the conditions: (1) The screen distribution S(TM) is totally umbilical (geodesic) and (2) the co-screen distribution $S(TM^{\bot})$ of M is a conformal Killing one.

ISOSPECTRAL MANIFOLDS WITH DIFFERENT LOCAL GEOMETRY

  • Gordon, Carolyn S.
    • Journal of the Korean Mathematical Society
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    • v.38 no.5
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    • pp.955-970
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    • 2001
  • Two compact Riemannian manifolds are said to be isospectral if the associated Laplace-Beltrami operators have the same eigenvalue spectrum. We describe a method, based on the used of Riemannian submersions, for constructing isospectral manifolds with different local geometry and survey examples constructed through this method.

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THEOREMS ON NULL-PATHS AND REDSHIFT

  • Wanas, M.I.;Morcos, A.B.
    • Journal of The Korean Astronomical Society
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    • v.46 no.3
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    • pp.97-102
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    • 2013
  • In the present work, we prove the validity of two theorems on null-paths in a version of absolute parallelismgeometry. A version of these theorems has been originally established and proved by Kermak, McCrea and Whittaker (KMW) in the context of Riemannian geometry. The importance of such theorems lies in their applications to derive a general formula for the redshift of spectral lines coming from distant objects. The formula derived in the present work can be applied to both cosmological and astrophysical redshifts. It takes into account the shifts resulting from gravitation, different motions of the source of photons, spin of the moving particle (photons) and the direction of the line of sight. It is shown that this formula cannot be derived in the context of Riemannian geometry, but it can be reduced to a formula given by KMW under certain conditions.

SHARP INEQUALITIES INVOLVING THE CHEN-RICCI INEQUALITY FOR SLANT RIEMANNIAN SUBMERSIONS

  • Mehmet Akif Akyol;Nergiz (Onen) Poyraz
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.5
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    • pp.1155-1179
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    • 2023
  • Main objective of the present paper is to establish Chen inequalities for slant Riemannian submersions in contact geometry. In this manner, we give some examples for slant Riemannian submersions and also investigate some curvature relations between the total space, the base space and fibers. Moreover, we establish Chen-Ricci inequalities on the vertical and the horizontal distributions for slant Riemannian submersions from Sasakian space forms.

A LIOUVILLE-TYPE THEOREM FOR COMPLETE RIEMANNIAN MANIFOLDS

  • Choi, Soon-Meen;Kwon, Jung-Hwan;Suh, Young-Jin
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.2
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    • pp.301-309
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    • 1998
  • The purpose of this paper is to give a theorem of Liouvilletype for complete Riemannian manifolds as an extension of the Theorem of Nishikawa [6].

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LIGHTLIKE HYPERSURFACES WITH TOTALLY UMBILICAL SCREEN DISTRIBUTIONS

  • Jin, Dae-Ho
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.3
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    • pp.409-416
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    • 2009
  • In this paper, we study the geometry of lightlike hypersurfaces of a semi-Riemannian manifold. We prove a classification theorem for lightlike hypersurfaces M with totally umbilical screen distributions of a semi-Riemannian space form.

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On the History of the Birth of Finsler Geometry at Göttingen (괴팅겐에서 핀슬러 기하가 탄생한 역사)

  • Won, Dae Yeon
    • Journal for History of Mathematics
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    • v.28 no.3
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    • pp.133-149
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    • 2015
  • Arrivals of Hilbert and Minkowski at $G\ddot{o}ttingen$ put mathematical science there in full flourish. They further extended its strong mathematical tradition of Gauss and Riemann. Though Riemann envisioned Finsler metric and gave an example of it in his inaugural lecture of 1854, Finsler geometry was officially named after Minkowski's academic grandson Finsler. His tool to generalize Riemannian geometry was the calculus of variations of which his advisor $Carath\acute{e}odory$ was a master. Another $G\ddot{o}ttingen$ graduate Busemann regraded Finsler geometry as a special case of geometry of metric spaces. He was a student of Courant who was a student of Hilbert. These figures all at $G\ddot{o}ttingen$ created and developed Finsler geometry in its early stages. In this paper, we investigate history of works on Finsler geometry contributed by these frontiers.

A NEW CLASS OF RIEMANNIAN METRICS ON TANGENT BUNDLE OF A RIEMANNIAN MANIFOLD

  • Baghban, Amir;Sababe, Saeed Hashemi
    • Communications of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.1255-1267
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    • 2020
  • The class of isotropic almost complex structures, J𝛿,𝜎, define a class of Riemannian metrics, g𝛿,𝜎, on the tangent bundle of a Riemannian manifold which are a generalization of the Sasaki metric. This paper characterizes the metrics g𝛿,0 using the geometry of tangent bundle. As a by-product, some integrability results will be reported for J𝛿,𝜎.