Invariants of Velocities, and Higher Order Grassmann Bundles
Abstract
An
(r,n)
-velocity is an
r
-jet with source at $0 \in \R^n$, and target in a manifold
Y
. An
(r,n)
-velocity is said to be regular, if it has a representative which is an immersion at $0 \in \R^{n}$. The manifold
T
r
n
Y
of
(r,n)
-velocities as well as its open,
L
r
n
-invariant, dense submanifold $\Imm T^{r}_{n}Y$ of regular
(r,n)
-velocities, are endowed with a natural action of the differential group
L
r
n
of invertible
r
-jets with source and target $0 \in \R^{n}$. In this paper, we describe all continuous,
L
r
n
-invariant, real-valued functions on
T
r
n
Y
and $\Imm T^{r}_{n}Y$. We find local bases of
L
r
n
-invariants on $\Imm T^{r}_{n}Y$ in an explicit, recurrent form. To this purpose, higher order Grassmann bundles are considered as the corresponding quotients $P^{r}_{n}Y = \Imm T^{r}_{n}Y/L^{r}_{n}$, and their basic properties are studied. We show that nontrivial
L
r
n
-invariants on $\Imm T^{r}_{n}Y$ cannot be continuously extended onto
T
r
n
Y
.