60 research outputs found
On the linearizability of 3-webs: end of controversy
There are two theories describing the linearizability of 3-webs: one is
developed in the article "On the linearizability of 3-webs" (Nonlinear analysis
47, (2001) pp.2643-2654) and another in the article "On the Blaschke conjecture
for 3-webs" (J. Geom. Anal. 16, 1 (2006), 69-115). Unfortunately, they cannot
be both correct because on an explicit 3-web W they contradict: the first
predicts that W is linearizable while the second states that W is not
linearizable. The essential question beyond this particular 3-web is: which
theory describes correctly the linearizability condition? In this paper we
present a very short proof, due to J.-P.~Dufour, that W is linearizable,
confirming the result of the first article
Projective Metrizability and Formal Integrability
The projective metrizability problem can be formulated as follows: under what
conditions the geodesics of a given spray coincide with the geodesics of some
Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective
metrizability problem for a spray in terms of a first-order partial
differential operator and a set of algebraic conditions on semi-basic
1-forms. We discuss the formal integrability of using two sufficient
conditions provided by Cartan-K\"ahler theorem. We prove in Theorem 4.2 that
the symbol of is involutive and hence one of the two conditions is always
satisfied. While discussing the second condition, in Theorem 4.3 we prove that
there is only one obstruction to the formal integrability of , and this
obstruction is due to the curvature tensor of the induced nonlinear connection.
When the curvature obstruction is satisfied, the projective metrizability
problem reduces to the discussion of the algebraic conditions, which as we show
are always satisfied in the analytic case. Based on these results, we recover
all classes of sprays that are known to be projectively metrizable: flat
sprays, isotropic sprays, and arbitrary sprays on 1- and 2-dimensional
manifolds. We provide examples of sprays that are projectively metrizable
without being Finsler metrizable
Metrizable isotropic second-order differential equations and Hilbert's fourth problem
It is well known that a system of homogeneous second-order ordinary
differential equations (spray) is necessarily isotropic in order to be
metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we
show that the isotropy condition, together with three other conditions on the
Jacobi endomorphism, characterize sprays that are metrizable by Finsler
functions of scalar flag curvature. The proof of Theorem 3.1 provides an
algorithm to construct the Finsler function of scalar flag curvature, in the
case when a given spray is metrizable. One condition of Theorem 3.1, regarding
the regularity of the sought after Finsler function, can be relaxed. By
relaxing this condition, we provide examples of sprays that are metrizable by
conic pseudo-Finsler functions as well as degenerate Finsler functions.
Hilbert's fourth problem asks to determine the Finsler functions with
rectilinear geodesics. A Finsler function that is a solution to Hilbert's
fourth problem is necessarily of constant or scalar flag curvature. Therefore,
we can use the conditions of [11, Theorem 4.1] and Theorem 3.1 to test when the
projective deformations of a flat spray, which are isotropic, are metrizable by
Finsler functions of constant or scalar flag curvature. We show how to use the
algorithms provided by the proofs of [11, Theorem 4.1] and Theorem 3.1 to
construct solutions to Hilbert's fourth problem.Comment: 15 page
Projective and Finsler metrizability: parameterization-rigidity of the geodesics
In this work we show that for the geodesic spray of a Finsler function
the most natural projective deformation leads to a non-Finsler metrizable spray, for almost every value of
. This result shows how rigid is the metrizablility
property with respect to certain reparameterizations of the geodesics. As a
consequence we obtain that the projective class of an arbitrary spray contains
infinitely many sprays that are not Finsler metrizable
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