On Projective Umbilics: a Geometric Invariant and an Index

Abstract : We define a geometric invariant and an index (+1 or −1) for projective umbilics of smooth surfaces. We prove that the sum of the indices of the projective umbilics inside a connected component H of the hyperbolic domain remains constant in any 1-parameter family of surfaces if the topological type of H does not change. We prove the same statement for any connected component E of the elliptic domain. We give formulas for the invariant and for the index which do not depend on any normal form.
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Submitted on : Friday, March 23, 2018 - 12:57:04 PM
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Ricardo Uribe-Vargas. On Projective Umbilics: a Geometric Invariant and an Index. Journal of Singularities, Worldwide Center of Mathematics, LLC, 2018, 17, pp.81 - 90. ⟨10.5427/jsing.2018.17e⟩. ⟨hal-01741660⟩

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