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Topological Classification of Morse Functions and Generalisations of Hilbert’s 16-th Problem

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Abstract

The topological structures of the generic smooth functions on a smooth manifold belong to the small quantity of the most fundamental objects of study both in pure and applied mathematics. The problem of their study has been formulated by A. Cayley in 1868, who required the classification of the possible configurations of the horizontal lines on the topographical maps of mountain regions, and created the first elements of what is called today ‘Morse Theory’ and ‘Catastrophes Theory’. In the paper we describe this problem, and in particular describe the classification of Morse functions on the 2 sphere and on the torus.

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Authors and Affiliations

  1. Steklov Mathematical Institut, Moscow, Russia

    Vladimir I. Arnold

  2. Université Paris Dauphine, Paris, France

    Vladimir I. Arnold

Authors
  1. Vladimir I. Arnold
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Correspondence to Vladimir I. Arnold.

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Arnold, V.I. Topological Classification of Morse Functions and Generalisations of Hilbert’s 16-th Problem. Math Phys Anal Geom 10, 227–236 (2007). https://doi.org/10.1007/s11040-007-9029-0

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  • DOI: https://doi.org/10.1007/s11040-007-9029-0

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