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Tangent lines, inflections, and vertices of closed curves. (English) Zbl 1295.53002

The authors present some important results in the study of closed curves in the Euclidean space \(\mathbb{R}^{3}\). The paper consists of eight sections. In the first section, some important notions are re-called concerning the theory of curves and some theorems are presented with inequalities on the number of singular points of a curve, which represent some of the main results of the paper. In the following sections, some other interesting results concerning closed curves are obtained. For the proofs of the main results, the author uses important results from projective geometry and geodesic theory. In the last section, some well-chosen examples are presented regarding the theory of closed curves. In conclusion, the paper is well written and presents some important results at the border between the theory of curves and projective geometry.

MSC:

53A04 Curves in Euclidean and related spaces
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
57R45 Singularities of differentiable mappings in differential topology
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
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References:

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