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Zap. Nauchn. Sem. POMI, 2018, Volume 471, Pages 211–224 (Mi znsl6633)  

This article is cited in 1 scientific paper (total in 1 paper)

On Morse index for geodesic lines on smooth surfaces imbedded in $\mathbb R^3$

M. M. Popov

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia

Abstract: The paper is devoted to calculation of Morse index on geodesic lines upon smooth surfaces embedded into 3D Euclidean space. The interest to this theme is called by the fact, that the wave field composed of the surface waves slides along the boundaries guided by the geodesic lines, which, generally speaking, give birth to numerous caustics. The same circumstance takes place in problems of the short-wave diffraction by 3D bodies in the shadowed part of the surface of the body, where the creeping waves arise.
We consider two types of geodesic flows upon the surface when they are generated by a point source and by an initial wave front, for instance, by the light-shadow boundary in the short-wave diffraction by a smooth convex body. We establish position of the points where geodesic lines meet caustics, i.e. focal points, and prove that all focal points are simple (not multiple) independently upon geometrical structure of the caustics arisen. Mathematical technique in use is based on complexification of geometrical spreading problem for the geodesics/rays tube.

Key words and phrases: geodesic lines, Fermat functional, equations in variations, geometrical spreading, Morse index.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-01-00529A


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English version:
Journal of Mathematical Sciences (New York), 2019, 243:5, 774–782

UDC: 517.9
Received: 07.09.2018

Citation: M. M. Popov, “On Morse index for geodesic lines on smooth surfaces imbedded in $\mathbb R^3$”, Mathematical problems in the theory of wave propagation. Part 48, Zap. Nauchn. Sem. POMI, 471, POMI, St. Petersburg, 2018, 211–224; J. Math. Sci. (N. Y.), 243:5 (2019), 774–782

Citation in format AMSBIB
\Bibitem{Pop18}
\by M.~M.~Popov
\paper On Morse index for geodesic lines on smooth surfaces imbedded in~$\mathbb R^3$
\inbook Mathematical problems in the theory of wave propagation. Part~48
\serial Zap. Nauchn. Sem. POMI
\yr 2018
\vol 471
\pages 211--224
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6633}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2019
\vol 243
\issue 5
\pages 774--782
\crossref{https://doi.org/10.1007/s10958-019-04577-3}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85075155555}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. M. Popov, “Novaya kontseptsiya poverkhnostnykh voln interferentsionnogo tipa dlya gladkikh strogo vypuklykh poverkhnostei, vlozhennykh v $\mathbb R^3$”, Matematicheskie voprosy teorii rasprostraneniya voln. 50, Posvyaschaetsya devyanostoletiyu Vasiliya Mikhailovicha BABIChA, Zap. nauchn. sem. POMI, 493, POMI, SPb., 2020, 301–313  mathnet
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