A horizontal endomorphism of the canonical superspray

Abstract

Giving up the homogeneity condition of a Lagrange superfunction, we prove that there is a unique horizontal endomorphism $h$ (nonlinear connection) on a supermanifold ${\mathcal{M}},$ such that $h$ is conservative and its torsion vanishes. There are several examples for nonhomogeneous Lagrangians such that this result is not true.

Keywords

• Horizontal endomorphism,Riemann-Finsler supermetric,Superspray

References

1. [1] W. Barthel, Nichtlineare zusammenh¨ange und deren holonomie gruppen, J. Reine Angew. Math. 212 (1963) 120-149.
2. [2] E. Cartan, Les espaces de Finsler, Hermann, Paris (1934).
3. [3] M. Crampin, On horizontal distributions on the tangent bundle of a differentiable manifold, J. Lond. Math. Soc., (2) 3, (1971) 178-182.
4. [4] B. DeWitt, Supermanifolds, (Cambridge: Cambridge University Press) 2nd edn, 1992.
5. [5] C. Ehresmann, Les connexions infinit´esimales dans un espace fibr´e diff´erentiable, Coll. Topologia, Bruxelles 29-55 (1950).
6. [6] J. Grifone, Structure presque-tangente et connexions. I. (French)Ann. Inst. Fourier (Grenoble) 22 (1972), no. 1, 287–334.
7. [7] J. Grifone, Structure presque-tangente et connexions. II. (French)Ann. Inst. Fourier (Grenoble) 22 (1972), no. 3, 291–338.
8. [8] L. A. Ibort ; G. Landi; J. Marn-Solano and G. Marmo, On the inverse problem of Lagrangian supermechanics, Internat. J. Modern Phys. A 8 (1993), no. 20, 3565–3576.

9. [9] A. Jadczyk and K. Pilch, Superspaces and supersymmetries, Comm. Math. Phys. 78 (1980) 373-390.
10. [10] A. Kawaguchi, Beziehung zwischen einer metrischen linearen Übertragung und einer nicht-metrischen in einem allgemeinen metrischen Rame, Proc. Akad. Wet. Amsterdam 40, (1937) 596-601.
11. [11] A. Kawaguchi, On the theory of non-linear connections II, theory of Minkowski spaces and of non-linear connections in a Finsler spaces,Tensor, New Ser. 6, 165-199 (1956).
12. [12] Y. Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys. 69 (2004), 61–87.
13. [13] M.M. Rezaii and E. Azizpour, On a superspray in Lagrange superspaces, Rep. Math. Phys. 56 (2005) 257-269.
14. [14] S. Vacaru and H. Dehnen, Locally anisotropic structures and nonlinear connections in Einstein and gauge gravity, Gen. Rel. Grav. 35 (2003) 209-250.
15. [15] S. I. Vacaru, Superstrings in higher order extensions of Finsler superspaces Nucl. Phys. B494 (1997) no. 3, 590-656.
16. [16] S. I. Vacaru, Interactions, strings and isotopies in higher order anisotropic superspaces, Hadronic Press, Palm Harbor, FL, USA, 1998.