• Title/Summary/Keyword: Invariant metrics

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LEFT INVARIANT LORENTZIAN METRICS AND CURVATURES ON NON-UNIMODULAR LIE GROUPS OF DIMENSION THREE

  • Ku Yong Ha;Jong Bum Lee
    • Journal of the Korean Mathematical Society
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    • v.60 no.1
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    • pp.143-165
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    • 2023
  • For each connected and simply connected three-dimensional non-unimodular Lie group, we classify the left invariant Lorentzian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the Ricci operator, the scalar curvature, and the sectional curvatures as functions of left invariant Lorentzian metrics on each of these groups. Our study is a continuation and extension of the previous studies done in [3] for Riemannian metrics and in [1] for Lorentzian metrics on unimodular Lie groups.

S-CURVATURE AND GEODESIC ORBIT PROPERTY OF INVARIANT (α1, α2)-METRICS ON SPHERES

  • Huihui, An;Zaili, Yan;Shaoxiang, Zhang
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.1
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    • pp.33-46
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    • 2023
  • Geodesic orbit spaces are homogeneous Finsler spaces whose geodesics are all orbits of one-parameter subgroups of isometries. Such Finsler spaces have vanishing S-curvature and hold the Bishop-Gromov volume comparison theorem. In this paper, we obtain a complete description of invariant (α1, α2)-metrics on spheres with vanishing S-curvature. Also, we give a description of invariant geodesic orbit (α1, α2)-metrics on spheres. We mainly show that a Sp(n + 1)-invariant (α1, α2)-metric on S4n+3 = Sp(n + 1)/Sp(n) is geodesic orbit with respect to Sp(n + 1) if and only if it is Sp(n + 1)Sp(1)-invariant. As an interesting consequence, we find infinitely many Finsler spheres with vanishing S-curvature which are not geodesic orbit spaces.

THE MODULI SPACES OF LORENTZIAN LEFT-INVARIANT METRICS ON THREE-DIMENSIONAL UNIMODULAR SIMPLY CONNECTED LIE GROUPS

  • Boucetta, Mohamed;Chakkar, Abdelmounaim
    • Journal of the Korean Mathematical Society
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    • v.59 no.4
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    • pp.651-684
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    • 2022
  • Let G be an arbitrary, connected, simply connected and unimodular Lie group of dimension 3. On the space 𝔐(G) of left-invariant Lorentzian metrics on G, there exists a natural action of the group Aut(G) of automorphisms of G, so it yields an equivalence relation ≃ on 𝔐(G), in the following way: h1 ≃ h2 ⇔ h2 = 𝜙*(h1) for some 𝜙 ∈ Aut(G). In this paper a procedure to compute the orbit space Aut(G)/𝔐(G) (so called moduli space of 𝔐(G)) is given.

Estimates of invariant metrics on some pseudoconvex domains in $C^N$

  • Cho, Sang-Hyun
    • Journal of the Korean Mathematical Society
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    • v.32 no.4
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    • pp.661-678
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    • 1995
  • In this paper we will estimate from above and below the values of the Bergman, Caratheodory and Kobayashi metrics for a vector X at z, where z is any point near a given point $z_0$ in the boundary of pseudoconvex domains in $C^n$.

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HOMOGENEOUS STRUCTURES ON FOUR-DIMENSIONAL LORENTZIAN DAMEK-RICCI SPACES

  • Assia Mostefaoui;Noura Sidhoumi
    • Communications of the Korean Mathematical Society
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    • v.38 no.1
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    • pp.195-203
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    • 2023
  • Special examples of harmonic manifolds that are not symmetric, proving that the conjecture posed by Lichnerowicz fails in the non-compact case have been intensively studied. We completely classify homogeneous structures on Damek-Ricci spaces equipped with the left invariant metric.

Improvement of ASIFT for Object Matching Based on Optimized Random Sampling

  • Phan, Dung;Kim, Soo Hyung;Na, In Seop
    • International Journal of Contents
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    • v.9 no.2
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    • pp.1-7
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    • 2013
  • This paper proposes an efficient matching algorithm based on ASIFT (Affine Scale-Invariant Feature Transform) which is fully invariant to affine transformation. In our approach, we proposed a method of reducing similar measure matching cost and the number of outliers. First, we combined the Manhattan and Chessboard metrics replacing the Euclidean metric by a linear combination for measuring the similarity of keypoints. These two metrics are simple but really efficient. Using our method the computation time for matching step was saved and also the number of correct matches was increased. By applying an Optimized Random Sampling Algorithm (ORSA), we can remove most of the outlier matches to make the result meaningful. This method was experimented on various combinations of affine transform. The experimental result shows that our method is superior to SIFT and ASIFT.

Scalar curvatures of invariant metrics

  • Park, Joon-Sik;Oh, Won-Tae
    • Journal of the Korean Mathematical Society
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    • v.31 no.4
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    • pp.629-632
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    • 1994
  • Let (M, g) be a Riemannian manifold. The relation between a (pointwise) conformal metric of the metric g and the scalar curvature of this new metrics is investigated by Kazdan, Warner and Schoen (cf. [1, 4]).

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ISOMETRY GOUP SO(1,2)

  • Kim, Sung-Sook;Shin, Joon-Kook
    • Communications of the Korean Mathematical Society
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    • v.11 no.4
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    • pp.1055-1059
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    • 1996
  • We characterize the left invariant Riemannian metrics on SO(1,2) which give rise to 3- or 4-dimensional isometry groups.

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R-CRITICAL WEYL STRUCTURES

  • Kim, Jong-Su
    • Journal of the Korean Mathematical Society
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    • v.39 no.2
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    • pp.193-203
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    • 2002
  • Weyl structure can be viewed as generalizations of Riemannian metrics. We study Weyl structures which are critical points of the squared L$^2$ norm functional of the full curvature tensor, defined on the space of Weyl structures on a compact 4-manifold. We find some relationship between these critical Weyl structures and the critical Riemannian metrics. Then in a search for homogeneous critical structures we study left-invariant metrics on some solv-manifolds and prove that they are not critical.