1,721,034 research outputs found
An intrinsic approach to the Geodesical Connectedness of Stationary Lorentzian Manifolds
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On the number of solutions for the two-point boundary value problem on Riemannian manifolds
No full textWe study the solutions joining two fixed points of a time-independent dynamical system on a Riemannian manifold (M, g) from an enumerative point of view. We prove a finiteness result for solutions joining two points p, q is an element of M that are non-conjugate in a suitable sense, under the assumption that (M, g) admits a non-trivial convex function. We discuss in some detail the notion of conjugacy induced by a general dynamical system on a Riemannian manifold. Using techniques of infinite dimensional Morse theory on Hilbert manifolds we also prove that, under generic circumstances, the number of solutions joining two fixed points is odd. We present some examples where our theory applie
On the relative category in the brake orbits problem
Full text linkIn this paper we show how the notion of the Lusternik–Schni- relmann relative category can be used to study a multiplicity problem for brake orbits in a potential well which is homeomorphic to the N-dimen- sional unit disk. The estimate of the relative category of the set of chords with endpoints on the (N − 1)-unit sphere was shown to the third author by Fadell and Husseini while he was visiting the University of Wisconsin at Madison
Convexity and the finiteness of the number of geodesics. Applications to the multiple-image effect
No full textWe discuss the relations between the notion of convexity in a Riemannian framework and finiteness of the number of geodesics between two fixed points. The results obtained can be applied to the study of lightlike geodesics joining a point and a timelike curve in a conformally static Lorentzian manifold. These geodesics are interpreted as light rays between a light source and an event of a relativistic spacetime; thus our results may be used for a mathematical description of the gravitational lensing effect
Convexity and the finiteness of the number of geodesics. Applications to the Gravitational Lensing Effect
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Gravitational lensing in general relativity via bifurcation theory
No full textIn a Lorentzian manifold, conjugate points along a lightlike are endpoints of homotopies of lightlike geodesics, up to first order infinitesimals. When studying phenomena on a very large scale in general relativity, like the gravitational lensing effect, the first order approximation is not a valid approach. In this paper we discuss a multiplicity result for light rays issuing from a light source and reaching an observer placed at a conjugate point of a lightlike geodesic using Krasnosel'skii–Rabinowitz bifurcation theory
Maslov index and Morse theory for the relativistic Lorentz force equation
No full textWe study the Jacobi equation for fixed endpoints solutions of the Lorentz force equation on a Lorentzian manifold. The flow of the Jacobi equation along each solution preserves the so-called twisted symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution. We introduce the notion of F-conjugate plane for each conjugate instant; the restriction of the spacetime metric to the F-conjugate plane is used to compute the Maslov index, which is given by a sort of algebraic count of the conjugate instants. For a stationary Lorentzian manifold and an exact electromagnetic field admitting a potential vector field preserving the flow of the Killing vector field, we introduce a constrained action functional having finite Morse index and whose critical points are fixed endpoints solution of the Lorentz force equation. We prove that the value of this Morse index equals the Maslov index and we prove the Morse relations for the solutions of the Lorentz force equation in a static spacetime
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