Geodesic
Projective Metrizability and Formal Integrability
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $P_1$ and a set of algebraic conditions on semi-basic $1$-forms. We discuss the formal integrability of $P_1$ using two sufficient conditions provided by Cartan–Kähler theorem. We prove in Theorem 4.2 that the symbol of $P_1$ 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 $P_1$, 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.
Keywords: sprays, projective metrizability, semi-basic forms, partial differential operators, formal integrability. 
@article{SIGMA_2011_7_a113,
     author = {Ioan Bucataru and Zolt\'an Muzsnay},
     title = {Projective {Metrizability} and {Formal} {Integrability}},
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     year = {2011},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a113/}
}
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Ioan Bucataru; Zoltán Muzsnay. Projective Metrizability and Formal Integrability. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a113/

[1] Álvarez Paiva J.C., “Symplectic geometry and Hilbert's fourth problem”, J. Differential Geom., 69 (2005), 353–378 | MR | Zbl

[2] Anderson I., Thompson G., “The inverse problem of the calculus of variations for ordinary differential equations”, Mem. Amer. Math. Soc., 98, no. 473, 1992 | MR | Zbl

[3] Antonelli P.L., Ingarden R.S., Matsumoto M., The theory of sprays and Finsler spaces with applications in physics and biology, Kluwer Academic Publisher, Dordrecht, 1993 | MR | Zbl

[4] Bucataru I., Constantinescu O.A., Dahl M.F., “A geometric setting for systems of ordinary differential equations”, Int. J. Geom. Methods Mod. Phys., 8 (2011), 1291–1327 ; arXiv: 1011.5799 | DOI

[5] Bucataru I., Dahl M.F., “Semi basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations”, J. Geom. Mech., 1 (2009), 159–180 ; arXiv: 0903.1169 | DOI | MR | Zbl

[6] Bucataru I., Muzsnay Z., Projective and Finsler metrizability: parameterization-rigidity of the geodesics, arXiv: 1108.4628

[7] Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffits P.A., Exterior differential systems, Mathematical Sciences Research Institute Publications, 18, Springer-Verlag, New York, 1991 | MR | Zbl

[8] Crampin M., “On the differential geometry of the Euler–Lagrange equation and the inverse problem of Lagrangian dynamics”, J. Phys. A: Math. Gen., 14 (1981), 2567–2575 | DOI | MR | Zbl

[9] Crampin M., “On the inverse problem for sprays”, Publ. Math. Debrecen, 70 (2007), 319–335 | MR | Zbl

[10] Crampin M., “Isotropic and $R$-flat sprays”, Houston J. Math., 33 (2007), 451–459 | MR | Zbl

[11] Crampin M., “Some remarks on the Finslerian version of Hilbert's fourth problem”, Houston J. Math., 37 (2011), 369–391 | MR | Zbl

[12] Darboux G., Leçons sur la theorie des surfaces, v. III, Gauthier-Villars, Paris, 1894 | Zbl

[13] Frölicher A., Nijenhuis A., “Theory ot vector-valued differential forms. I. Derivations in the graded ring of differential forms”, Nederl. Akad. Wet. Proc. Ser. A, 59 (1956), 338–359 | MR

[14] Grifone J., “Structure presque-tangente et connexions. I”, Ann. Inst. Fourier (Grenoble), 22 (1972), 287–334 | DOI | MR | Zbl

[15] Grifone J., Muzsnay Z., Variational principles for second-order differential equations. Application of the Spencer theory to characterize variational sprays, World Scientific Publishing Co., Inc., River Edge, NJ, 2000 | DOI | MR | Zbl

[16] Hamel G., “Über die Geometrien, in denen die Geraden die Kürzesten sind”, Math. Ann., 57 (1903), 231–264 | DOI | MR | Zbl

[17] Klein J., Voutier A., “Formes extérieures génératrices de sprays”, Ann. Inst. Fourier (Grenoble), 18 (1968), 241–260 | DOI | MR | Zbl

[18] Kolár I., Michor P.W., Slovak J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993 | MR

[19] Krupková O., Prince G.E., “Second order ordinary differential equations in jet bundles and the inverse problem of the calculus of variations”, Handbook of Global Analysis, eds. D. Krupka and D.J. Saunders, Elsevier Sci. B.V., Amsterdam, 2007, 837–904 | DOI | MR

[20] de León M., Rodrigues P.R., Methods of differential geometry in analytical mechanics, North-Holland Mathematics Studies, 158, North-Holland Publishing Co., Amsterdam, 1989 | MR | Zbl

[21] Lovas R.L., “A note on Finsler–Minkowski norms”, Houston J. Math., 33 (2007), 701–707 | MR | Zbl

[22] Matsumoto M., “Every path space of dimension two is projectively related to a Finsler space”, Open Syst. Inf. Dyn., 3 (1995), 291–303 | DOI | Zbl

[23] Matsumoto M., Foundations of Finsler geometry and special Finsler spaces, Kaiseisha Press, Shigaken, 1986 | MR | Zbl

[24] Miron R., Anastasiei M., The geometry of Lagrange spaces: theory and applications, Fundamental Theories of Physics, 59, Kluwer Academic Publishers Group, Dordrecht, 1994 | MR | Zbl

[25] Morandi G., Ferrario C., Lo Vecchio G., Marmo G., Rubano C., “The inverse problem in the calculus of variations and the geometry of the tangent bundle”, Phys. Rep., 188 (1990), 147–284 | DOI | MR | Zbl

[26] Muzsnay Z., “The Euler–Lagrange PDE and Finsler metrizability”, Houston J. Math., 32 (2006), 79–98 ; arXiv: math.DG/0602383 | MR | Zbl

[27] Rapcsák A., “Die Bestimmung der Grundfunktionen projektiv-ebener metrischer Räume”, Publ. Math. Debrecen, 9 (1962), 164–167 | MR | Zbl

[28] Sarlet W., “The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics”, J. Phys. A: Math. Gen., 15 (1982), 1503–1517 | DOI | MR | Zbl

[29] Shen Z., Differential geometry of spray and Finsler spaces, Kluwer Academic Publishers, Dordrecht, 2001 | MR

[30] Szilasi J., “Calculus along the tangent bundle projection and projective metrizability”, Differential Geometry and its Applications, World Sci. Publ., Hackensack, NJ, 2008, 539–558 | DOI | MR | Zbl

[31] Szilasi J., Vattamány S., “On the Finsler-metrizabilities of spray manifolds”, Period. Math. Hungar., 44 (2002), 81–100 | DOI | MR | Zbl

[32] Yang G., “Some classes of sprays in projective spray geometry”, Differential Geom. Appl., 29 (2011), 606–614 | DOI | MR | Zbl