Hilbert forms for a Finsler metrizable projective class of sprays

• Published in: Differential geometry and its applications Vol. 31; no. 1; pp. 63 - 79
• Main Authors: Crampin, M., Mestdag, T., Saunders, D.J.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2013
• Subjects: Almost Grassmann structure; Differential geometry; Equivalence; Finsler function; Geodesic path; Hilbert form; Jacobi field; Mathematics; Path geometry; Path space; Projective equivalence; Projective metrizability; Spray; Sprayers; Sprays; Totally-geodesic submanifold; Finsler function; Hilbert form; Path geometry; Projective metrizability; Almost Grassmann structure; Totally-geodesic submanifold; Geodesic path; Spray; Jacobi field; 53C60; Projective equivalence; Path space
• ISSN: 0926-2245; 1872-6984
• DOI: 10.1016/j.difgeo.2012.10.012

The projective Finsler metrizability problem deals with the question whether a projective-equivalence class of sprays is the geodesic class of a (locally- or globally-defined) Finsler function. In this paper we use Hilbert-type forms to state a number of different ways of specifying necessary and sufficient conditions for this to be the case, and we show that they are equivalent. We also address several related issues of interest including path spaces, Jacobi fields, totally-geodesic submanifolds of a spray space, and the equivalence of path geometries and projective-equivalence classes of sprays.