Relative and Absolute Differential Invariants for Conformal Curves

Gloria Marí Beffa

Abstract: In this paper we classify all vector relative differential invariants with Jacobian weight for the conformal action of O$(n+1,1)$ on parametrized curves in ${\Bbb R}^{n}$. 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 in Fialkow, A., {\it The Conformal Theory of Curves}, ``Transactions of the AMS'' {\bf 51} (1942), 435--56. As a corollary, we obtain the most general formula for evolutions of curves in ${\Bbb R}^{n}$ invariant under the conformal action of the group.

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