Invariants of Velocities, and Higher Order Grassmann Bundles

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 $T^{r}_{n}Y$.