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Invariants of velocities and higher-order Grassmann bundles. (English) Zbl 0898.53013

The paper deals with an easily formulated problem: what are the functions of the \(r\)th order \(n\)-dimensional velocities, invariant under the obvious right action by the jet group \(L^r_n\)? The velocities were studied intensively already by Ehresmann and, intuitively, also the answer is quite natural: the orbits of the action containing regular velocities form exactly the bundles of contact elements (also called Grassmann bundles) and so the continuous invariant functions of the regular velocities are given by any continuous functions on the contact elements. On the other hand, the neighborhoods of the classes of the most singular velocities in the space of the orbits are that big that the only invariant continuous functions defined globally are coming from functions on the underlying manifolds. The paper consists of the proof of the latter claim.

MSC:

53A55 Differential invariants (local theory), geometric objects
57S25 Groups acting on specific manifolds
58A20 Jets in global analysis
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