BUNCH CONNECTIONS

George Virsik

Key words: Path connection, semi-smooth, $r$-bunch,
bunch connection, bloop.

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


Abstract: A path connection in a principal bundle $P(M,G)$ is
a lifting of paths in $M$ onto paths in $P$. It is a well known fact
that such liftings which are independent of reparametrisation,
equivariant and infinitesimal of order one are in a one-to-one
correspondence with ordinary connections in $P$. Even if
$r$-th order jets are used, a path connection can be reasonably
associated only with an $r$-th order connection which is the
Ehresmann prolongation of a single first order connection.
In order to make a similar relation hold for arbitrary (non-holonomic)
$r$-th order connections, one needs to lift $r$-bunches
rather than paths. Here an $r$-bunch in $M$ is a map $I^r\to M$,
not necessarily continuous, with a certain semi-smoothness condition.
In particular, the bunch connection associated with an
$r$-th order connection that is the product of
$r$ first order connections is investigated.