AN INFINITE DIMENSIONAL MOTIVATION IN HIGHER ORDER GEOMETRY

Ivan  Kol\'{a}\v{r}

1991 Math. Subject Classification: 58A20, 53C05.

Key words: Fiber jet, bundle of smooth maps, Weil bundle, exchange
map, iterated absolute differentiation.

Abstract: Given two fiber bundles $Y_1$ and $Y_2$ over the same base, we
start with a survey of some  geometric constructions on the space
$\Cal F(Y_1, Y_2)$ of all smooth maps of a fiber of $Y_1$ into the
fiber of $Y_2$ over the same base point. In particular, the theory
of Weil bundles can be extended from finite dimensional manifolds
to $\Cal F(Y_1,Y_2)$. A systematic development of such an approach
yields  some new operations on the jet  prolongations of vertical
Weil bundles and on similar objects. The applications to the
theory of connections are discussed.