Differential and Integral Equations

An intrinsic approach to Ljusternik-Schnirelman theory for light rays on Lorentzian manifolds

Flavia Antonacci and Paolo Piccione

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In this paper it is proven the existence of light--like geodesics joining an event $p$ and a time--like vertical curve $\gamma$ of a Lorentzian manifold $\mathcal{M}$ endowed with a Universal Time Function $T$, under a certain compactness condition. Moreover, it is developed a Ljusternik--Schnirelman theory for light rays, using which it is shown that, if the topology of $\mathcal{M}$ satisfies a non--triviality condition, then there are multiple light rays joining $p$ with $\gamma$. The results are obtained under intrinsic assumptions on the manifold $\mathcal{M}$, that do not involve the coefficients of the Lorentzian metric

Article information

Differential Integral Equations, Volume 12, Number 4 (1999), 521-562.

First available in Project Euclid: 29 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58E10: Applications to the theory of geodesics (problems in one independent variable)
Secondary: 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Antonacci, Flavia; Piccione, Paolo. An intrinsic approach to Ljusternik-Schnirelman theory for light rays on Lorentzian manifolds. Differential Integral Equations 12 (1999), no. 4, 521--562. https://projecteuclid.org/euclid.die/1367267006

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