Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 114, 22 pages      arXiv:1105.2142

Projective Metrizability and Formal Integrability

Ioan Bucataru a and Zoltán Muzsnay b
a) Faculty of Mathematics, Al.I.Cuza University, B-dul Carol 11, Iasi, 700506, Romania
b) Institute of Mathematics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary

Received August 25, 2011, in final form December 08, 2011; Published online December 12, 2011

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 P1 and a set of algebraic conditions on semi-basic 1-forms. We discuss the formal integrability of P1 using two sufficient conditions provided by Cartan-Kähler theorem. We prove in Theorem 4.2 that the symbol of P1 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 P1, 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.

Key words: sprays; projective metrizability; semi-basic forms; partial differential operators; formal integrability.

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