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Keywords:
geodesic; shell of a curve; affine connection; (pseudo-)Riemannian metric; projective equivalence
geodesic; shell of a curve; affine connection; (pseudo-)Riemannian metric; projective equivalence
Summary:
We determine in $\mathbb {R}^n$ the form of curves $C$ corresponding to strictly monotone functions as well as the components of affine connections $\nabla $ for which any image of $C$ under a compact-free group $\Omega $ of affinities containing the translation group is a geodesic with respect to $\nabla $. Special attention is paid to the case that $\Omega $ contains many dilatations or that $C$ is a curve in $\mathbb {R}^3$. If $C$ is a curve in $\mathbb {R}^3$ and $\Omega $ is the translation group then we calculate not only the components of the curvature and the Weyl tensor but we also decide when $\nabla $ yields a flat or metrizable space and compute the corresponding metric tensor.
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