• Title/Summary/Keyword: Finsler geometry

Search Result 22, Processing Time 0.024 seconds

On the History of the Birth of Finsler Geometry at Göttingen (괴팅겐에서 핀슬러 기하가 탄생한 역사)

  • Won, Dae Yeon
    • Journal for History of Mathematics
    • /
    • v.28 no.3
    • /
    • pp.133-149
    • /
    • 2015
  • Arrivals of Hilbert and Minkowski at $G\ddot{o}ttingen$ put mathematical science there in full flourish. They further extended its strong mathematical tradition of Gauss and Riemann. Though Riemann envisioned Finsler metric and gave an example of it in his inaugural lecture of 1854, Finsler geometry was officially named after Minkowski's academic grandson Finsler. His tool to generalize Riemannian geometry was the calculus of variations of which his advisor $Carath\acute{e}odory$ was a master. Another $G\ddot{o}ttingen$ graduate Busemann regraded Finsler geometry as a special case of geometry of metric spaces. He was a student of Courant who was a student of Hilbert. These figures all at $G\ddot{o}ttingen$ created and developed Finsler geometry in its early stages. In this paper, we investigate history of works on Finsler geometry contributed by these frontiers.

On the history of 60 years of Japanese School of Finsler Geometry (일본 핀슬러 기하학파의 60년 역사)

  • Won, Dae Yeon
    • Journal for History of Mathematics
    • /
    • v.34 no.3
    • /
    • pp.89-111
    • /
    • 2021
  • This paper is a continuation of the study on the history of the Japanese school of Finsler geometry. We had studied on the birth of Japanese school of Finsler geometry. In this paper, we find out what motivated Japanese to embrace Finsler geometry and we collect the history and analyze trends of Japanese school of Finsler geometry since its founding by M. Matsumoto.

On the history of the establishment of the Hungarian Debrecen School of Finsler geometry after L. Berwald (베어왈트에 의한 헝가리 데브레첸 핀슬러 기하학파의 형성의 역사)

  • Won, Dae Yeon
    • Journal for History of Mathematics
    • /
    • v.31 no.1
    • /
    • pp.37-51
    • /
    • 2018
  • In this paper, our main concern is the historical development of the Finsler geometry in Debrecen, Hungary initiated by L. Berwald. First we look into the research trend in Berwald's days affected by the $G{\ddot{o}}ttingen$ mathematicians from C. Gauss and downward. Then we study how he was motivated to concentrate on the then completely new research area, Finsler geometry. Finally we examine the course of establishing Hungarian Debrecen school of Finsler geometry via the scholars including O. Varga, A. $Rapcs{\acute{a}}k$, L. $Tam{\acute{a}}ssy$ all deeply affected by Berwald after his settlement in Debrecen, Hungary.

On the beginning of the formation of Japanese School of Finsler Geometry (일본 핀슬러 기하학파 형성의 시작에 관하여)

  • Won, Dae Yeon
    • Journal for History of Mathematics
    • /
    • v.34 no.2
    • /
    • pp.55-74
    • /
    • 2021
  • Matsumoto Makoto is regarded as founding father of the Japanese school of Finsler geometry because he established the Japanese Society of Finsler Geometry in 1968 and organized the Symposium every year since then. In this paper, we investigate how Matsumoto initiated the study of this topic leaping over geographical limit and how Yano Kentaro and Kawaguchi Akitsugu had affected Matsumoto in the formation of the Japanese school of Finsler geometry. We also take a view of the role of É. Cartan who invented the concept of the connection in early 20th century in this regard.

ON THE SYNGE'S THEOREM FOR COMPLEX FINSLER MANIFOLDS

  • Won, Dae-Yeon
    • Bulletin of the Korean Mathematical Society
    • /
    • v.41 no.1
    • /
    • pp.137-145
    • /
    • 2004
  • In [13], we developed a theory of complex Finsler manifolds to investigate the global geometry of complex Finsler manifolds. There we proved a version of Bonnet-Myers' theorem for complex Finsler manifolds with a certain condition on the Finsler metric which is a generalization of the Kahler condition for the Hermitian metric. In this paper, we show that if the holomorphic sectional curvature of M is ${\geq}\;c^2\;>\;0$, then M is simply connected. This is a generalization of the Synge's theorem in the Riemannian geometry and the Tsukamoto's theorem for Kahler manifolds. The main point of the proof lies in how we can circumvent the convex neighborhood theorem in the Riemannian geometry. A second variation formula of arc length for complex Finsler manifolds is also derived.

MEDICAL IMAGE ANALYSIS USING HIGH ANGULAR RESOLUTION DIFFUSION IMAGING OF SIXTH ORDER TENSOR

  • K.S. DEEPAK;S.T. AVEESH
    • Journal of applied mathematics & informatics
    • /
    • v.41 no.3
    • /
    • pp.603-613
    • /
    • 2023
  • In this paper, the concept of geodesic centered tractography is explored for diffusion tensor imaging (DTI). In DTI, where geodesics has been tracked and the inverse of the fourth-order diffusion tensor is inured to determine the diversity. Specifically, we investigated geodesic tractography technique for High Angular Resolution Diffusion Imaging (HARDI). Riemannian geometry can be extended to a direction-dependent metric using Finsler geometry. Euler Lagrange geodesic calculations have been derived by Finsler geometry, which is expressed as HARDI in sixth order tensor.

THEORY OF HYPERSURFACES OF A FINSLER SPACE WITH THE GENERALIZED SQUARE METRIC

  • SONIA RANI;VINOD KUMAR;MOHAMMAD RAFEE
    • Journal of applied mathematics & informatics
    • /
    • v.42 no.4
    • /
    • pp.879-897
    • /
    • 2024
  • The emergence of generalized square metrics in Finsler geometry can be attributed to various classification concerning (𝛼, 𝛽)-metrics. They have excellent geometric properties in Finsler geometry. Within the scope of this research paper, we have conducted an investigation into the generalized square metric denoted as $F(x,y)=\frac{[{\alpha}(x,y)+{\beta}(x,y)]^{n+1}}{[{\alpha}(x,y)]^n}$, focusing specifically on its application to the Finslerian hypersurface. Furthermore, the classification and existence of first, second, and third kind of hyperplanes of the Finsler manifold has been established.

COMPARISON THEOREMS IN FINSLER GEOMETRY WITH WEIGHTED CURVATURE BOUNDS AND RELATED RESULTS

  • Wu, Bing-Ye
    • Journal of the Korean Mathematical Society
    • /
    • v.52 no.3
    • /
    • pp.603-624
    • /
    • 2015
  • We first extend the notions of weighted curvatures, including the weighted flag curvature and the weighted Ricci curvature, for a Finsler manifold with given volume form. Then we establish some basic comparison theorems for Finsler manifolds with various weighted curvature bounds. As applications, we obtain some McKean type theorems for the first eigenvalue of Finsler manifolds, some results on weighted curvature and fundamental group for Finsler manifolds, as well as an estimation of Gromov simplicial norms for reversible Finsler manifolds.

INTRINSIC THEORY OF Cv-REDUCIBILITY IN FINSLER GEOMETRY

  • Salah Gomaa Elgendi;Amr Soleiman
    • Communications of the Korean Mathematical Society
    • /
    • v.39 no.1
    • /
    • pp.187-199
    • /
    • 2024
  • In the present paper, following the pullback approach to Finsler geometry, we study intrinsically the Cv-reducible and generalized Cv-reducible Finsler spaces. Precisely, we introduce a coordinate-free formulation of these manifolds. Then, we prove that a Finsler manifold is Cv-reducible if and only if it is C-reducible and satisfies the 𝕋-condition. We study the generalized Cv-reducible Finsler manifold with a scalar π-form 𝔸. We show that a Finsler manifold (M, L) is generalized Cv-reducible with 𝔸 if and only if it is C-reducible and 𝕋 = 𝔸. Moreover, we prove that a Landsberg generalized Cv-reducible Finsler manifold with a scalar π-form 𝔸 is Berwaldian. Finally, we consider a special Cv-reducible Finsler manifold and conclude that a Finsler manifold is a special Cv-reducible if and only if it is special semi-C-reducible with vanishing 𝕋-tensor.