Araştırma Makalesi
BibTex RIS Kaynak Göster

On Metrics and Linear Connections on Lines

Yıl 2025, Cilt: 18 Sayı: 1, 60 - 85, 24.04.2025

Jun-ichi Inoguchi

Öz

We discuss linear connections and conformal Riemannian metricson the real line.

Anahtar Kelimeler

Linear connection Hilbert distance; geodesic HMC surface harmonic map Hessian manifold

Kaynakça

  • Bercu, G., Corcodel, C., Postolache, M.: Iterative geometric structures, Int. J. Geom. Methods Mod. Phys. 7 (7), 1103-1114 (2010).
  • Bobenko, A. I.: Surfaces in terms of 2 by 2 matrices. Old and new integrable cases, Harmonic Maps and Integrable Systems, Aspects of Math. 83, Vieweg, 83-127 (1994).
  • Bobenko, A., Eitner, U., Kitaev, A.: Surfaces with harmonic inverse mean curvature and Painlevé equations, Geom. Dedicata 68 (2), 187-227 (1997).
  • Crasmareanu, M.: Weighted Riemannian 1-manifolds for classical orthogonal polynomials and their heat kernel, Anal. Math. Phys. 5 (4), 373–389 (2015).
  • Fujioka, A.: Surfaces with harmonic inverse mean curvature in space forms, Proc. Amer. Math. Soc. 127 (10), 3021-3025 (1999).
  • Fujioka, A., Inoguchi, J.: On some generalisations of constant mean curvature surfaces, Lobachevskii J. Math. 3, 73-95 (1999).
  • Fujioka, A., Inoguchi, J.: Spacelike surfaces with harmonic inverse mean curvature, J. Math. Sci. Univ. Tokyo 7 (4), 657-698 (2000).
  • Fujioka, A., Inoguchi, J.: Timelike Bonnet surfaces in Lorentzian space forms, Differential Geom. Appl. 18 (1), 103-111 (2003).
  • Fujioka, A., Inoguchi, J.: Timelike surfaces with harmonic inverse mean curvature, Surveys on Geometry and Integrable Systems, Advanced Studies in Pure Mathematics 51, 113-141 (2018).
  • Furuhata, H., Inoguchi, J., Kobayashi, S.-P.: A characterization of the alpha-connections on the statistical manifold of normal distributions, Inf. Geom. 4 (1), 177-188 (2021).
  • Furuhata, H., Kurose, T.: Hessian manifolds of nonpositive constant Hessian sectional curvature, Tôhoku Math. J. (2) 65 (1), 31-42 (2013). 31–42.
  • H. Furuhata, H., Ueno, R.: A variation problem for mappings between statistical manifolds, Results in Mathematics, 80 (57), (2025).
  • Grigor’yan, A.: Heat Kernel and Analysis on Manifolds, AMS/IP Stud. Adv. Math. 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009.
  • Goldman, W. M.: Flat affine, projective and conformal structures on manifolds: A historical perspective, Geometry in History, Springer, 515-552, (2019).
  • Hilbert, D.: Ueber die gerade Linie als kürzeste Verbindung zweier Punkte, Math. Ann. 46, 91-96 (1895).
  • Inoguchi, J.: On the statistical Lie groups of normal distributions, Information Geometry, 7 (2), 441-447 (2024).
  • Inoguchi, J., Ohno, Y.: Homogeneous statistical manifolds, arXiv:2408.01647v1 [math.DG]
  • Kito, H.: On Hessian structures on the Euclidean space and the hyperbolic space, Osaka J. Math. 36 (1), 51-62 (1999).
  • Kobayashi, O., Wada, M.: Circular geometry and the Schwarzian, Far East J. Math. Sci, Special Volume Part III, 335-363 (2000).
  • Kobayashi, S.: Projective structures and invariant distances, (Japanese), Sugaku ¯ 34 (3), 211-221 (1982).
  • Kobayashi, S.: Projectively invariant distances for affine and projective structures, Differential Geometry, Warsaw 1979, Banach Cent. Publ. 12, 127-152 (1984).
  • Molitor, M.: One-dimensional exponential families with constant Hessian sectional curvature, Inf. Geo. 5, 511-530 (2022).
  • Nomizu, K., Sasaki, T.: Globally defined linear connections on the real line and the circle, Tôhoku Math. J. (2) 51 (2), 205-212 (1999).
  • Park, J.-S.: Harmonic inner automorphisms of compact connected semisimple Lie groups, Tôhoku Math. J. (2) 42 (1), 83-91 (1990).
  • Shima, H.: Hessian manifolds of constant Hessian sectional curvature, J. Math. Soc. Japan 47, 735-753 (1995).
  • Shima, H.: The geometry of Hessian structures, World Scientific, Hackensack, NJ, 2007.
  • Goldman, W. M.: Projective geometry on manifolds, Lecture Notes for Mathematics 748B, Spring 1988, University of Maryland.
  • Kobayashi, O.: On a theorem of N. H. Kuiper, (Japanese), Geometry and Analysis 2023, Fukuoka University, 20 pages.
  • Kobayashi, S.-P., Ohno, Y.: On a constant curvature statistical manifold, Inf. Geom. 5 (1), 31-46 (2022).
  • Kuiper, N. H.: Locally projective spaces of dimension one, Michigan Math. J. 2 (2), 95-97 (1953/1954).
  • Osipov, P.: Locally conformally Hessian and statistical manifolds, J. Geom. Phys. 193, Paper No. 104989 (2023).

Yıl 2025, Cilt: 18 Sayı: 1, 60 - 85, 24.04.2025

Jun-ichi Inoguchi

Öz

Kaynakça

  • Bercu, G., Corcodel, C., Postolache, M.: Iterative geometric structures, Int. J. Geom. Methods Mod. Phys. 7 (7), 1103-1114 (2010).
  • Bobenko, A. I.: Surfaces in terms of 2 by 2 matrices. Old and new integrable cases, Harmonic Maps and Integrable Systems, Aspects of Math. 83, Vieweg, 83-127 (1994).
  • Bobenko, A., Eitner, U., Kitaev, A.: Surfaces with harmonic inverse mean curvature and Painlevé equations, Geom. Dedicata 68 (2), 187-227 (1997).
  • Crasmareanu, M.: Weighted Riemannian 1-manifolds for classical orthogonal polynomials and their heat kernel, Anal. Math. Phys. 5 (4), 373–389 (2015).
  • Fujioka, A.: Surfaces with harmonic inverse mean curvature in space forms, Proc. Amer. Math. Soc. 127 (10), 3021-3025 (1999).
  • Fujioka, A., Inoguchi, J.: On some generalisations of constant mean curvature surfaces, Lobachevskii J. Math. 3, 73-95 (1999).
  • Fujioka, A., Inoguchi, J.: Spacelike surfaces with harmonic inverse mean curvature, J. Math. Sci. Univ. Tokyo 7 (4), 657-698 (2000).
  • Fujioka, A., Inoguchi, J.: Timelike Bonnet surfaces in Lorentzian space forms, Differential Geom. Appl. 18 (1), 103-111 (2003).
  • Fujioka, A., Inoguchi, J.: Timelike surfaces with harmonic inverse mean curvature, Surveys on Geometry and Integrable Systems, Advanced Studies in Pure Mathematics 51, 113-141 (2018).
  • Furuhata, H., Inoguchi, J., Kobayashi, S.-P.: A characterization of the alpha-connections on the statistical manifold of normal distributions, Inf. Geom. 4 (1), 177-188 (2021).
  • Furuhata, H., Kurose, T.: Hessian manifolds of nonpositive constant Hessian sectional curvature, Tôhoku Math. J. (2) 65 (1), 31-42 (2013). 31–42.
  • H. Furuhata, H., Ueno, R.: A variation problem for mappings between statistical manifolds, Results in Mathematics, 80 (57), (2025).
  • Grigor’yan, A.: Heat Kernel and Analysis on Manifolds, AMS/IP Stud. Adv. Math. 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009.
  • Goldman, W. M.: Flat affine, projective and conformal structures on manifolds: A historical perspective, Geometry in History, Springer, 515-552, (2019).
  • Hilbert, D.: Ueber die gerade Linie als kürzeste Verbindung zweier Punkte, Math. Ann. 46, 91-96 (1895).
  • Inoguchi, J.: On the statistical Lie groups of normal distributions, Information Geometry, 7 (2), 441-447 (2024).
  • Inoguchi, J., Ohno, Y.: Homogeneous statistical manifolds, arXiv:2408.01647v1 [math.DG]
  • Kito, H.: On Hessian structures on the Euclidean space and the hyperbolic space, Osaka J. Math. 36 (1), 51-62 (1999).
  • Kobayashi, O., Wada, M.: Circular geometry and the Schwarzian, Far East J. Math. Sci, Special Volume Part III, 335-363 (2000).
  • Kobayashi, S.: Projective structures and invariant distances, (Japanese), Sugaku ¯ 34 (3), 211-221 (1982).
  • Kobayashi, S.: Projectively invariant distances for affine and projective structures, Differential Geometry, Warsaw 1979, Banach Cent. Publ. 12, 127-152 (1984).
  • Molitor, M.: One-dimensional exponential families with constant Hessian sectional curvature, Inf. Geo. 5, 511-530 (2022).
  • Nomizu, K., Sasaki, T.: Globally defined linear connections on the real line and the circle, Tôhoku Math. J. (2) 51 (2), 205-212 (1999).
  • Park, J.-S.: Harmonic inner automorphisms of compact connected semisimple Lie groups, Tôhoku Math. J. (2) 42 (1), 83-91 (1990).
  • Shima, H.: Hessian manifolds of constant Hessian sectional curvature, J. Math. Soc. Japan 47, 735-753 (1995).
  • Shima, H.: The geometry of Hessian structures, World Scientific, Hackensack, NJ, 2007.
  • Goldman, W. M.: Projective geometry on manifolds, Lecture Notes for Mathematics 748B, Spring 1988, University of Maryland.
  • Kobayashi, O.: On a theorem of N. H. Kuiper, (Japanese), Geometry and Analysis 2023, Fukuoka University, 20 pages.
  • Kobayashi, S.-P., Ohno, Y.: On a constant curvature statistical manifold, Inf. Geom. 5 (1), 31-46 (2022).
  • Kuiper, N. H.: Locally projective spaces of dimension one, Michigan Math. J. 2 (2), 95-97 (1953/1954).
  • Osipov, P.: Locally conformally Hessian and statistical manifolds, J. Geom. Phys. 193, Paper No. 104989 (2023).
Toplam 31 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Cebirsel ve Diferansiyel Geometri
Bölüm Araştırma Makalesi
Yazarlar

Jun-ichi Inoguchi 0000-0002-6584-5739

Erken Görünüm Tarihi 20 Nisan 2025
Yayımlanma Tarihi 24 Nisan 2025
Gönderilme Tarihi 7 Haziran 2024
Kabul Tarihi 27 Ekim 2024
Yayımlandığı Sayı Yıl 2025 Cilt: 18 Sayı: 1

Kaynak Göster

APA Inoguchi, J.-i. (2025). On Metrics and Linear Connections on Lines. International Electronic Journal of Geometry, 18(1), 60-85.
AMA Inoguchi Ji. On Metrics and Linear Connections on Lines. Int. Electron. J. Geom. Nisan 2025;18(1):60-85.
Chicago Inoguchi, Jun-ichi. “On Metrics and Linear Connections on Lines”. International Electronic Journal of Geometry 18, sy. 1 (Nisan 2025): 60-85.
EndNote Inoguchi J-i (01 Nisan 2025) On Metrics and Linear Connections on Lines. International Electronic Journal of Geometry 18 1 60–85.
IEEE J.-i. Inoguchi, “On Metrics and Linear Connections on Lines”, Int. Electron. J. Geom., c. 18, sy. 1, ss. 60–85, 2025.
ISNAD Inoguchi, Jun-ichi. “On Metrics and Linear Connections on Lines”. International Electronic Journal of Geometry 18/1 (Nisan 2025), 60-85.
JAMA Inoguchi J-i. On Metrics and Linear Connections on Lines. Int. Electron. J. Geom. 2025;18:60–85.
MLA Inoguchi, Jun-ichi. “On Metrics and Linear Connections on Lines”. International Electronic Journal of Geometry, c. 18, sy. 1, 2025, ss. 60-85.
Vancouver Inoguchi J-i. On Metrics and Linear Connections on Lines. Int. Electron. J. Geom. 2025;18(1):60-85.
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