Abstract
The solution space of a constant coefficient ODE gives rise to a natural real analytic curve in Euclidean space. We give necessary and sufficient conditions on the ODE to ensure that this curve is a proper embedding of infinite length or has finite total first curvature. If all the roots of the associated characteristic polynomial are simple, we give a uniform upper bound for the total first curvature and show the optimal uniform upper bound must grow at least linearly with the order $n$ of the ODE. We then examine the case where multiple roots are permitted. We present several examples illustrating that a curve can have finite total first curvature for positive/negative time and infinite total first curvature for negative/positive time as well as illustrating that other possibilities may occur.
Citation
P. GILKEY. C. Y. KIM. H. MATSUDA. J. H. PARK. S. YOROZU. "Non-closed curves in ℝn with finite total first curvature arising from the solutions of an ODE." Hokkaido Math. J. 46 (1) 119 - 139, February 2017. https://doi.org/10.14492/hokmj/1498788099
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