On Projective Umbilics: a Geometric Invariant and an Index
Copyright policy )On Projective Umbilics: a Geometric Invariant and an Index
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.
13 pages, 7 figures
- Collège de France France
- University of Burgundy France
index, Mathematics - Differential Geometry, Theory of singularities and catastrophe theory, [MATH] Mathematics [math], cross-ratio, Mathematics - Geometric Topology, parabolic curve, flecnodal curve, FOS: Mathematics, Contact manifolds (general theory), surface, differential geometry, projective umbilic, quadratic point, Projective differential geometry, Differential invariants (local theory), geometric objects, Geometric Topology (math.GT), singularity, 53A20, 53A55, 53D10, 57R45, 58K05, Critical points of functions and mappings on manifolds, Differential Geometry (math.DG), invariant, [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Singularities of differentiable mappings in differential topology
index, Mathematics - Differential Geometry, Theory of singularities and catastrophe theory, [MATH] Mathematics [math], cross-ratio, Mathematics - Geometric Topology, parabolic curve, flecnodal curve, FOS: Mathematics, Contact manifolds (general theory), surface, differential geometry, projective umbilic, quadratic point, Projective differential geometry, Differential invariants (local theory), geometric objects, Geometric Topology (math.GT), singularity, 53A20, 53A55, 53D10, 57R45, 58K05, Critical points of functions and mappings on manifolds, Differential Geometry (math.DG), invariant, [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Singularities of differentiable mappings in differential topology
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