Journal of the Mathematical Society of Japan

A conjecture in relation to Loewner's conjecture

Naoya ANDO

Full-text: Open access


Let f be a smooth function of two variables x , y and for each positive integer n , let d n f be a symmetric tensor field of type ( 0 , n ) defined by d n f : = i = 0 n n i x n - i y i f d x n - i d y i and 𝒟 ˜ d n f a finitely many-valued one-dimensional distribution obtained from d n f : for example, 𝒟 ˜ d 1 f is the one-dimensional distribution defined by the gradient vector field of f ; 𝒟 ˜ d 2 f consists of two one-dimensional distributions obtained from one-dimensional eigenspaces of Hessian of f . In the present paper, we shall study the behavior of 𝒟 ˜ d n f around its isolated singularity in ways which appear in [1]--[4]. In particular, we shall introduce and study a conjecture which asserts that the index of an isolated singularity with respect to 𝒟 ˜ d n f is not more than one.

Article information

J. Math. Soc. Japan, Volume 57, Number 1 (2005), 1-20.

First available in Project Euclid: 13 October 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37E35: Flows on surfaces
Secondary: 53A05: Surfaces in Euclidean space 53B25: Local submanifolds [See also 53C40]

Loewner's conjecture the index conjecture Carathéodory's conjecture symmetric tensor field critical direction umbilical point many-valued one-dimensional distribution index


ANDO, Naoya. A conjecture in relation to Loewner's conjecture. J. Math. Soc. Japan 57 (2005), no. 1, 1--20. doi:10.2969/jmsj/1160745810.

Export citation


  • N. Ando, An isolated umbilical point of the graph of a homogeneous polynomial, Geom. Dedicata, 82 (2000), 115–137.
  • N. Ando, The behavior of the principal distributions around an isolated umbilical point, J. Math. Soc. Japan, 53 (2001), 237–260.
  • N. Ando, The behavior of the principal distributions on the graph of a homogeneous polynomial, Tohoku Math. J., 54 (2002), 163–177.
  • N. Ando, The behavior of the principal distributions on a real-analytic surface, J. Math. Soc. Japan, 56 (2004), 201–214.
  • L. Bates, A weak counterexample to the Carathéodory's conjecture, Differential Geom. Appl., 15 (2001), 79–80.
  • C. Gutierrez and F. Sanchez-Bringas, Planer vector field versions of Carathéodory's and Loewner's conjectures, Publ. Mat., 41 (1997), no.,1, 169–179.
  • H. Hopf, Differential geometry in the large, Lecture Notes in Math., vol.,1000, Springer, Berlin-NewYork, 1989.
  • T. Klotz, On G. Bol's proof of Carathéodory's conjecture, Comm. Pure Appl. Math., 12 (1959), 277–311.
  • H. Scherbel, A new proof of Hamburger's index theorem on umbilical points, Dissertation, ETH, Zürich, No.,10281, 1994.
  • B. Smyth and F. Xavier, A sharp geometric estimate for the index of an umbilic on a smooth surface, Bull. London Math. Soc., 24 (1992), 176–180.
  • B. Smyth and F. Xavier, Real solvability of the equation $\partial^2_{\overline{z}} \omega =\rho \fg$ and the topology of isolated umbilics, J. Geom. Anal., 8 (1998), 655–671.
  • B. Smyth and F. Xavier, Eigenvalue estimates and the index of Hessian fields, Bull. London Math. Soc., 33 (2001), 109–112.
  • C. J. Titus, A proof of a conjecture of Loewner and of the conjecture of Carathéodory on umbilic points, Acta Math., 131 (1973), 43–77.