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arXiv · 1207.4720

The Equivalence Problem of Curves in a Riemannian Manifold

M. Castrillon Lopez·V. Fernandez Mateos·J. Munoz Masque
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Abstract

The equivalence problem of curves with values in a Riemannian manifold, is solved. The domain of validity of Frenet's theorem is shown to be the spaces of constant curvature. For a general Riemannian manifold new invariants must thus be added. There are two important generic classes of curves; namely, Frenet curves and a new class, called curves "in normal position". They coincide in dimensions $\leq 4$ only. A sharp bound for asymptotic stability of differential invariants is obtained, the complete systems of invariants are characterized, and a procedure of generation is presented. Different classes of examples (specially in low dimensions) are analyzed in detail.

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BibTeXRIS

M. Castrillon Lopez, V. Fernandez Mateos, J. Munoz Masque. 2012-07-19. The Equivalence Problem of Curves in a Riemannian Manifold. https://arxiv.org/abs/1207.4720

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