• Title/Summary/Keyword: hyperbolic space

Search Result 141, Processing Time 0.03 seconds

MODULI OF SELF-DUAL METRICS ON COMPLEX HYPERBOLIC MANIFOLDS

  • Kim, Jaeman
    • Bulletin of the Korean Mathematical Society
    • /
    • v.39 no.1
    • /
    • pp.133-140
    • /
    • 2002
  • On compact complex hyperbolic manifolds of complex dimension two, we show that the dimension of the space of infinitesimal deformations of self-dual conformal structures is smaller than that of the deformation obstruction space and that every self-dual metric with covariantly constant Ricci tensor must be a standard one upto rescalings and diffeomorphisms.

A CHARACTERIZATION OF HOROSPHERES AND GEODESIC HYPERSPHERES IN A COMPLEX HYPERBOLIC SPACE IN TERMS OF RICCI TENSORS

  • Ahn, Seong-Soo
    • Bulletin of the Korean Mathematical Society
    • /
    • v.35 no.3
    • /
    • pp.503-514
    • /
    • 1998
  • We want to treat this problem for real hypersurfaces in a complex hyperbolic space $J_n(C)$. Thus it seems to be natural to consider some problems concerned with the estimation of the Ricci tensor for real hypersurfaces in $H_n(C)$. In this paper we will find a new tensorial formula concerned with the Ricci tensor and give it a characterization of horospheres and geodesic hyperspheres in a complex hyperbolic space $H_n(C)$.

  • PDF

CERTAIN CURVATURE CONDITIONS OF REAL HYPERSURFACES IN A COMPLEX HYPERBOLIC SPACE

  • Kim, Hyang Sook;Pak, Jin Suk
    • Communications of the Korean Mathematical Society
    • /
    • v.30 no.2
    • /
    • pp.131-142
    • /
    • 2015
  • The purpose of this paper is to study real hypersurfaces immersed in a complex hyperbolic space $CH^n$ and especially to investigate certain curvature conditions for such real hypersurfaces to be the model hypersurfaces in classification theorem (said to be Theorem M-R) given by Montiel and Romero ([4]) in Section 3.

FUNDAMENTAL TONE OF COMPLETE WEAKLY STABLE CONSTANT MEAN CURVATURE HYPERSURFACES IN HYPERBOLIC SPACE

  • Min, Sung-Hong
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.34 no.4
    • /
    • pp.369-378
    • /
    • 2021
  • In this paper, we give an upper bound for the fundamental tone of stable constant mean curvature hypersurfaces in hyperbolic space. Let M be an n-dimensional complete non-compact constant mean curvature hypersurface with finite L2-norm of the traceless second fundamental form. If M is weakly stable, then λ1(M) is bounded above by n2 + O(n2+s) for arbitrary s > 0.

IDEAL RIGHT-ANGLED PENTAGONS IN HYPERBOLIC 4-SPACE

  • Kim, Youngju;Tan, Ser Peow
    • Journal of the Korean Mathematical Society
    • /
    • v.56 no.4
    • /
    • pp.1131-1158
    • /
    • 2019
  • An ideal right-angled pentagon in hyperbolic 4-space ${\mathbb{H}}^4$ is a sequence of oriented geodesics ($L_1,{\ldots},L_5$) such that $L_i$ intersects $L_{i+1},i=1,{\ldots},4$, perpendicularly in ${\mathbb{H}}^4$ and the initial point of $L_1$ coincides with the endpoint of $L_5$ in the boundary at infinity ${\partial}{\mathbb{H}}^4$. We study the geometry of such pentagons and the various possible augmentations and prove identities for the associated quaternion half side lengths as well as other geometrically defined invariants of the configurations. As applications we look at two-generator groups ${\langle}A,B{\rangle}$ of isometries acting on hyperbolic 4-space such that A is parabolic, while B and AB are loxodromic.

IDEAL RIGHT-ANGLED PENTAGONS IN HYPERBOLIC 4-SPACE

  • Kim, Youngju;Tan, Ser Peow
    • Journal of the Korean Mathematical Society
    • /
    • v.56 no.3
    • /
    • pp.595-622
    • /
    • 2019
  • An ideal right-angled pentagon in hyperbolic 4-space ${\mathbb{H}}^4$ is a sequence of oriented geodesics ($L_1,{\ldots},L_5$) such that Li intersects $L_{i+1},\;i=1,\;{\ldots},\;4$, perpendicularly in ${\mathbb{H}}^4$ and the initial point of $L_1$ coincides with the endpoint of $L_5$ in the boundary at infinity ${\partial}{\mathbb{H}}^4$. We study the geometry of such pentagons and the various possible augmentations and prove identities for the associated quaternion half side lengths as well as other geometrically defined invariants of the configurations. As applications we look at two-generator groups ${\langle}A,B{\rangle}$ of isometries acting on hyperbolic 4-space such that A is parabolic, while B and AB are loxodromic.

GENERALIZED 𝛼-NONEXPANSIVE MAPPINGS IN HYPERBOLIC SPACES

  • Kim, Jong Kyu;Dashputre, Samir;Padmavati, Padmavati;Sakure, Kavita
    • Nonlinear Functional Analysis and Applications
    • /
    • v.27 no.3
    • /
    • pp.449-469
    • /
    • 2022
  • This paper deals with the new iterative algorithm for approximating the fixed point of generalized 𝛼-nonexpansive mappings in a hyperbolic space. We show that the proposed iterative algorithm is faster than all of Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur and Piri iteration processes for contractive mappings in a Banach space. We also establish some weak and strong convergence theorems for generalized 𝛼-nonexpansive mappings in hyperbolic space. The examples and numerical results are provided in this paper for supporting our main results.

On characterizations of real hypersurfaces of type B in a complex hyperbolic space

  • Ahn, Seong-Soo;Suh, Young-Jin
    • Journal of the Korean Mathematical Society
    • /
    • v.32 no.3
    • /
    • pp.471-482
    • /
    • 1995
  • A complex n-dimensional Kaehlerian manifold of constant holomorphic sectional curvature c is called a comples space form, which is denoted by $M_n(c)$. A complete and simply connected complex space form consists of a complex projective space $P_nC$, a complex Euclidean space $C^n$ or a complex hyperbolic space $H_nC$, according as c > 0, c = 0 or c < 0. The induced almost contact metric structure of a real hypersurface M of $M_n(c)$ is denoted by $(\phi, \zeta, \eta, g)$.

  • PDF