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Bull. Lond. Math. Soc., 2012, Volume 44, Issue 4, Pages 637–641 (Mi blms3)  

A topological proof of the Arnold four cusps theorem

V. A. Vassilievab

a Steklov Mathematical Institute, 8 Gubkina str., Moscow 119991, Russia
b Higher School of Economics

Abstract: The Arnold theorem (generalizing a consideration by Jacobi) states that on a generic Riemannian surface, which is sufficiently close to a sphere, the kth caustic of a generic point has at least four semi-cubical vertices. We prove this fact by the methods of the Morse theory; in particular, we replace the previous analytical condition of the ‘sufficient closeness to the sphere’ by a geometric one, which probably is considerably less restrictive.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation NSh-8462.2010.1
This work was supported by the programme ‘Leading scientific schools’, grant No. NSh-8462.2010.1.


DOI: https://doi.org/10.1112/blms/bdr119


Bibliographic databases:

MSC: 58E10
Received: 08.06.2011
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