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In this paper a Weil approach to quasijets is discussed. For given manifolds $M$ and $N$, a quasijet with source $x\in M$ and target $y\in N$ is a mapping $T^r_xM\to T^r_yN$ which is a vector homomorphism for each one of the $r$ vector bundle structures of the iterated tangent bundle $T^r$ [{\it A. Dekr\'et}, Casopis Pest. Mat. 111, No. 4, 345-352 (1986; Zbl 0611.58004)]. \par Let us denote by $QJ^r(M,N)$ the bundle of quasijets from $M$ to $N$; the space $\widetilde J^r(M,N)$ of non-holonomic $r$-jets from $M$ to $N$ is embeded into $QJ^r(M,N)$. On the other hand, the bundle $QT^r_mN$ of $(m,r)$-quasivelocities of $N$ is defined to be $QJ^r_0({\bold R}^m,N)$; then, $QT^r_m$ is a product preserving functor and so a Weil functor $T^{{\bold Q}^r_m}$ where ${\bold Q}^r_m$ is the Weil algebra $QT^r_m{\bold R}$ [see {\it I. Kol\'ar, P. Michor} and {\it J. Slov\'ak}, `Natural operations in differential geometry' (Springer-Verlag, Berlin) (1993; Zbl 0782.53013)]; next, t! (English) |
