The spray and antispray theory in the subspaces of Miron's $Osc^kM$

Irena Čomić, Jelena Stojanov, Gabrijela Grujić

The spray theory in $Osc^kM$ was introduced by R. Miron and Gh. Atanasiu in [7, 8] R. Miron with coauthors in [6, 9, 10] gave a comprehensive theory of higher order geometry and the spray theory. In [2] I. Comic reported the relation between $J$ structure, Liouville vector fields and the $S$-vector field as a more general basis than used by R. Miron and with the different variable $y^{k\alpha}=\frac{d^kx^\alpha}{dt^k}$ ($y^{k\alpha}=\frac1{k!}\frac{d^kx^\alpha}{dt^k}$ in Miron’s papers). Here, the adapted basis is changed in such a way that the mentioned relations have a new, simpler and more elegant form. The combinatorial aspect was also used and the notion of antispray is introduced. Using the specially adapted bases the spray and antispray theories in the subspaces of $Osc^kM$1 were established.