• Title/Summary/Keyword: Riemannian

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LINEAR WEINGARTEN HYPERSURFACES IN RIEMANNIAN SPACE FORMS

  • Chao, Xiaoli;Wang, Peijun
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.567-577
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    • 2014
  • In this note, we generalize the weak maximum principle in [4] to the case of complete linear Weingarten hypersurface in Riemannian space form $\mathbb{M}^{n+1}(c)$ (c = 1, 0,-1), and apply it to estimate the norm of the total umbilicity tensor. Furthermore, we will study the linear Weingarten hypersurface in $\mathbb{S}^{n+1}(1)$ with the aid of this weak maximum principle and extend the rigidity results in Li, Suh, Wei [13] and Shu [15] to the case of complete hypersurface.

POLYNOMIAL GROWTH HARMONIC MAPS ON COMPLETE RIEMANNIAN MANIFOLDS

  • Lee, Yong-Hah
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.3
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    • pp.521-540
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    • 2004
  • In this paper, we give a sharp estimate on the cardinality of the set generating the convex hull containing the image of harmonic maps with polynomial growth rate on a certain class of manifolds into a Cartan-Hadamard manifold with sectional curvature bounded by two negative constants. We also describe the asymptotic behavior of harmonic maps on a complete Riemannian manifold into a regular ball in terms of massive subsets, in the case when the space of bounded harmonic functions on the manifold is finite dimensional.

A CHARACTERIZATION OF SPACE FORMS

  • Kim, Dong-Soo;Kim, Young-Ho
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.757-767
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    • 1998
  • For a Riemannian manifold $(M^n, g)$ we consider the space $V(M^n, g)$ of all smooth functions on $M^n$ whose Hessian is proportional to the metric tensor $g$. It is well-known that if $M^n$ is a space form then $V(M^n)$ is of dimension n+2. In this paper, conversely, we prove that if $V(M^n)$ is of dimension $\ge{n+1}$, then $M^n$ is a Riemannian space form.

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PROJECTIVELY FLAT FINSLER SPACES WITH CERTAIN (α, β)-METRICS

  • Park, Hong-Suh;Park, Ha-Yong;Kim, Byung-Doo;Choi, Eun-Seo
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.4
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    • pp.649-661
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    • 2003
  • The ($\alpha,\;\beta$)-metric is a Finsler metric which is constructed from a Riemannian metric $\alpha$ and a differential 1-form $\beta$. In this paper, we discuss the projective flatness of Finsler spaces with certain ($\alpha,\;\beta$)-metrics ([5]) in a locally Minkowski space.

ON TRANSVERSALLY HARMONIC MAPS OF FOLIATED RIEMANNIAN MANIFOLDS

  • Jung, Min-Joo;Jung, Seoung-Dal
    • Journal of the Korean Mathematical Society
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    • v.49 no.5
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    • pp.977-991
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    • 2012
  • Let (M,F) and (M',F') be two foliated Riemannian manifolds with M compact. If the transversal Ricci curvature of F is nonnegative and the transversal sectional curvature of F' is nonpositive, then any transversally harmonic map ${\phi}:(M,F){\rightarrow}(M^{\prime},F^{\prime})$ is transversally totally geodesic. In addition, if the transversal Ricci curvature is positive at some point, then ${\phi}$ is transversally constant.

A NOTE ON GENERALIZED DIRAC EIGENVALUES FOR SPLIT HOLONOMY AND TORSION

  • Agricola, Ilka;Kim, Hwajeong
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.6
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    • pp.1579-1589
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    • 2014
  • We study the Dirac spectrum on compact Riemannian spin manifolds M equipped with a metric connection ${\nabla}$ with skew torsion $T{\in}{\Lambda}^3M$ in the situation where the tangent bundle splits under the holonomy of ${\nabla}$ and the torsion of ${\nabla}$ is of 'split' type. We prove an optimal lower bound for the first eigenvalue of the Dirac operator with torsion that generalizes Friedrich's classical Riemannian estimate.

4-DIMENSIONAL CRITICAL WEYL STRUCTURES

  • Kim, Jong-Su
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.3
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    • pp.551-564
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    • 2001
  • We view Weyl structures as generalizations of Riemannian metrics and study the critical points of geometric functional which involve scalar curvature, defined on the space of Weyl structures on a closed 4-manifold. The main goal here is to provide a framework to analyze critical Weyl structures by defining functionals, discussing function spaces and writing down basic formulas for the equations of critical points.

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ON THE SPECTRAL GEOMETRY FOR THE JACOBI OPERATORS OF HARMONIC MAPS INTO PRODUCT MANIFOLDS

  • Kang, Tae-Ho;Ki, U-Hang;Pak, Jin-Suk
    • Journal of the Korean Mathematical Society
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    • v.34 no.2
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    • pp.483-500
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    • 1997
  • We investigate the geometric properties reflected by the spectra of the Jacobi operator of a harmonic map when the target manifold is a Riemannian product manifold or a Kaehlerian product manifold. And also we study the spectral characterization of Riemannian sumersions when the target manifold is $S^n \times S^n$ or $CP^n \times CP^n$.

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ON THE STRUCTURE OF MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF NON-NEGATIVE CURVATURE

  • Yun, Gab-Jin;Kim, Dong-Ho
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.6
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    • pp.1213-1219
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    • 2009
  • Let M$^n$ be a complete oriented non-compact minimally immersed submanifold in a complete Riemannian manifold N$^{n+p}$ of nonnegative curvature. We prove that if M is super-stable, then there are no non-trivial L$^2$ harmonic one forms on M. This is a generalization of the main result in [8].