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Journal of Lie Theory 13 (2003), No. 1, 213--245
Copyright Heldermann Verlag 2003

Relative and Absolute Differential Invariants for Conformal Curves

Gloria Mari Beffa
Dept. of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A.

We classify all vector relative differential invariants with Jacobian weight for the conformal action of  O(n+1, 1) on parametrized curves in  Rn.  We then write a generating set of independent conformal differential invariants, for both parametrized and unparametrized curves, as simple combinations of the relative invariants.  We also find an invariant frame for unparametrized curves via a Gram-Schmidt procedure.  The invariants of unparametrized curves correspond to the ones found by A. Fialkow ["The conformal theory of curves", Transactions of the AMS 51 (1942) 435--456].  As a corollary, we obtain the most general formula for evolutions of curves in  R invariant under the conformal action of the group.

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