• Title/Summary/Keyword: Riemannian map

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REMARKS ON METALLIC MAPS BETWEEN METALLIC RIEMANNIAN MANIFOLDS AND CONSTANCY OF CERTAIN MAPS

  • Akyol, Mehmet Akif
    • Honam Mathematical Journal
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    • v.41 no.2
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    • pp.343-356
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    • 2019
  • In this paper, we introduce metallic maps between metallic Riemannian manifolds, provide an example and obtain certain conditions for such maps to be totally geodesic. We also give a sufficient condition for a map between metallic Riemannian manifolds to be harmonic map. Then we investigate the constancy of certain maps between metallic Riemannian manifolds and various manifolds by imposing the holomorphic-like condition. Moreover, we check the reverse case and show that some such maps are constant if there is a condition for this.

REMARKS ON LEVI HARMONICITY OF CONTACT SEMI-RIEMANNIAN MANIFOLDS

  • Perrone, Domenico
    • Journal of the Korean Mathematical Society
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    • v.51 no.5
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    • pp.881-895
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    • 2014
  • In a recent paper [10] we introduced the notion of Levi harmonic map f from an almost contact semi-Riemannian manifold (M, ${\varphi}$, ${\xi}$, ${\eta}$, g) into a semi-Riemannian manifold $M^{\prime}$. In particular, we compute the tension field ${\tau}_H(f)$ for a CR map f between two almost contact semi-Riemannian manifolds satisfying the so-called ${\varphi}$-condition, where $H=Ker({\eta})$ is the Levi distribution. In the present paper we show that the condition (A) of Rawnsley [17] is related to the ${\varphi}$-condition. Then, we compute the tension field ${\tau}_H(f)$ for a CR map between two arbitrary almost contact semi-Riemannian manifolds, and we study the concept of Levi pluriharmonicity. Moreover, we study the harmonicity on quasicosymplectic manifolds.

DIFFERENTIAL GEOMETRIC PROPERTIES ON THE HEISENBERG GROUP

  • Park, Joon-Sik
    • Journal of the Korean Mathematical Society
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    • v.53 no.5
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    • pp.1149-1165
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    • 2016
  • In this paper, we show that there exists no left invariant Riemannian metric h on the Heisenberg group H such that (H, h) is a symmetric Riemannian manifold, and there does not exist any H-invariant metric $\bar{h}$ on the Heisenberg manifold $H/{\Gamma}$ such that the Riemannian connection on ($H/{\Gamma},\bar{h}$) is a Yang-Mills connection. Moreover, we get necessary and sufficient conditions for a group homomorphism of (SU(2), g) with an arbitrarily given left invariant metric g into (H, h) with an arbitrarily given left invariant metric h to be a harmonic and an affine map, and get the totality of harmonic maps of (SU(2), g) into H with a left invariant metric, and then show the fact that any affine map of (SU(2), g) into H, equipped with a properly given left invariant metric on H, does not exist.

Construction of a complete negatively curved singular riemannian foliation

  • Haruo Kitahara;Pak, Hong-Kyung
    • Journal of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.609-614
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    • 1995
  • Let (M, g) be a complete Riemannian manifold and G be a closed (connected) subgroup of the group of isometries of M. Then the union ${\MM}$ of all principal orbits is an open dense subset of M and the quotient map ${\MM} \longrightarrow {\BB} := {\MM}/G$ becomes a Riemannian submersion for the restriction of g to ${\MM}$ which gives the quotient metric on ${\BB}$. Namely, B is a singular (complete) Riemannian space such that $\partialB$ consists of non-principal orbits.

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GRADIENT ESTIMATE OF HEAT EQUATION FOR HARMONIC MAP ON NONCOMPACT MANIFOLDS

  • Kim, Hyun-Jung
    • Journal of applied mathematics & informatics
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    • v.28 no.5_6
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    • pp.1461-1466
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    • 2010
  • aSuppose that (M, g) is a complete Riemannian manifold with Ricci curvature bounded below by -K < 0 and (N, $\bar{b}$) is a complete Riemannian manifold with sectional curvature bounded above by a constant $\mu$ > 0. Let u : $M{\times}[0,\;{\infty}]{\rightarrow}B_{\tau}(p)$ is a heat equation for harmonic map. We estimate the energy density of u.

AN ENERGY DENSITY ESTIMATE OF HEAT EQUATION FOR HARMONIC MAP

  • Kim, Hyun-Jung
    • The Pure and Applied Mathematics
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    • v.18 no.1
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    • pp.79-86
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    • 2011
  • Suppose that (M,g) is a complete and noncompact Riemannian mani-fold with Ricci curvature bounded below by $-K{\leq}0$ and (N, $\bar{g}$) is a complete Riemannian manifold with nonpositive sectional curvature. Let u : $M{\times}[0,{\infty}){\rightarrow}N$ be the solution of a heat equation for harmonic map with a bounded image. We estimate the energy density of u.

CLAIRAUT POINTWISE SLANT RIEMANNIAN SUBMERSION FROM NEARLY KÄHLER MANIFOLDS

  • Gauree Shanker;Ankit Yadav
    • Honam Mathematical Journal
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    • v.45 no.1
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    • pp.109-122
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    • 2023
  • In the present article, we introduce pointwise slant Riemannian submersion from nearly Kähler manifold to Riemannian manifold. We established the conditions for fibers to be totally geodesic. We also find necessary and sufficient conditions for pointwise slant submersion 𝜑 to be a harmonic and totally geodesic. Further, we study clairaut pointwise slant Riemannian submersion from nearly Kähler manifold to Riemannian manifold. We derive the clairaut conditions for 𝜑 such that 𝜑 is a clairaut map. Finally, one example is constructed which demonstrates existence of clairaut pointwise slant submersion from nearly Kähler manifold to Riemannian manifold.

EXISTENCE OF HOMOTOPIC HARMONIC MAPS INTO METRIC SPACE OF NONPOSITIVE CURVATURE

  • Jeon, Myung-Jin
    • Communications of the Korean Mathematical Society
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    • v.10 no.4
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    • pp.931-941
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    • 1995
  • The definitions and techniques, which deals with homotopic harmonic maps from a compact Riemannian manifold into a compact metric space, developed by N. J. Korevaar and R. M. Schoen [7] can be applied to more general situations. In this paper, we prove that for a complicated domain, possibly noncompact Riemannian manifold with infinitely generated fundamental group, the existence of homotopic harmonic maps can be proved if the initial map is simple in some sense.

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LIOUVILLE TYPE THEOREM FOR p-HARMONIC MAPS II

  • Jung, Seoung Dal
    • Communications of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.155-161
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    • 2014
  • Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that $Ric^M{\geq}-\frac{4(p-1)}{p^2}{\mu}_0$ at all $x{\in}M$ and Vol(M) is infinite, where ${\mu}_0$ > 0 is the infimum of the spectrum of the Laplacian acting on $L^2$-functions on M. Then any p-harmonic map ${\phi}:M{\rightarrow}N$ of finite p-energy is constant Also, we study Liouville type theorem for p-harmonic morphism.