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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
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Necessary and sufficient curvature condition for a regular planar curve to be simple and closed
The Four-vertex theorem dictates that the curvature function needs to have at least two local minima and maxima. The converse of the four-vertex theorem tells us that there's at least one parametrizat …