Abstract
We determine in ℝn the form of curves C corresponding to strictly monotone functions as well as the components of affine connections ▿ for which any image of C under a compact-free group Ω of affinities containing the translation group is a geodesic with respect to ▿. Special attention is paid to the case that Ω contains many dilatations or that C is a curve in ℝ3. If C is a curve in ℝ3 and Ω is the translation group then we calculate not only the components of the curvature and the Weyl tensor but we also decide when ▿ yields a flat or metrizable space and compute the corresponding metric tensor.
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The authors are supported by the grant from Grant Agency of Czech Republic GA ČR P201/11/0356, and by the Department Mathematik der Universität Erlangen-Nürnberg.
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Mikeš, J., Strambach, K. Shells of monotone curves. Czech Math J 65, 677–699 (2015). https://doi.org/10.1007/s10587-015-0202-5
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DOI: https://doi.org/10.1007/s10587-015-0202-5