Affinely invariant symmetry sets

Giblin, Peter

doi:10.4064/bc82-0-5

Cited by 11 publications

(8 citation statements)

References 16 publications

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“…Finally, let A(x, κ) be the area of the planar region bounded by M and a chord, considered as a function of a point x on the chord and a variable κ locating one of the endpoints of the chord on the curve. Then, A(x, κ) is a generating family for E 1/2 (M) [3,13]. Below we generalize this notion to every λ-equidistant of any Lagrangian submanifold.…”

Section: Theorem 28 the Set Gcs(m) Contains The Wigner Caustic Of Mmentioning

confidence: 99%

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Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

Domitrz

Rios

2013

Geom Dedicata

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We study the global centre symmetry set (GCS) of a smooth closed submanifold M m ⊂ R n , n ≤ 2m. The GCS includes both the centre symmetry set defined by Janeczko (Geometria Dedicata 60:9-16, 1996) and the Wigner caustic defined by Berry (Philos Trans R Soc Lond A 287:237-271, 1977) . The definition of GCS(M) uses the concept of an affinedp i ∧ dq i ), we present generating families for singularities of E λ (L) and prove that the caustic of any simple stable Lagrangian singularity in a 4m-dimensional Lagrangian fibre bundle is realizable as the germ of an affine equidistant of some L ⊂ R 2m . We characterize the criminant part of GCS(L) in terms of bitangent hyperplanes to L. Then, after presenting the appropriate equivalence relation to be used in this Lagrangian case, we classify the affine-Lagrangian stable singularities of GCS(L). In particular we show that, already for a smooth closed convex curve L ⊂ R 2 , many singularities of GCS(L) which are affine stable are not affine-Lagrangian stable.

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Section: Theorem 28 the Set Gcs(m) Contains The Wigner Caustic Of Mmentioning

confidence: 99%

“…First, in Theorem 7.1 we collect results on the GCS of convex curves in non-symplectic plane, [3,[9][10][11][12][13]16], and we obtain in Theorem 7.2 a new inequality on the number of cusps of the centre symmetry set and the Wigner caustic. Pictures illustrate these results.…”

Section: We Call This New Set the Global Centre Symmetry Set Of M Andmentioning

confidence: 99%

Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

Domitrz

Rios

2013

Geom Dedicata

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“…Geometrically, the set x(S) consists of the midpoints of chords connecting α(s) and β(t) with parallel tangents. In the theory of symmetry sets of planar curves, this set is called area evolute, or midpoint parallel tangent locus [4,5]. In case α(s) and β(t) are parameterizations of the same planar curve without parallel tangents, the set x(S) is just the curve itself.…”

Section: Singularity Setmentioning

confidence: 99%

“…Thus it coincides with a well-known symmetry set associated with the curves α and β, the area evolute, also called midpoint parallel tangent locus [4,5].…”

Section: Introductionmentioning

confidence: 99%

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A geometric representation of improper indefinite affine spheres with singularities

Craizer

Teixeira

Silva³

2011

J. Geom.

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Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefinite improper affine sphere in R 3 is the generalized distance of a pair of planar curves. Considering this representation, the singularity set of the improper affine sphere corresponds to the area evolute of the pair of curves, and this fact allows us to describe a clear geometric picture of the former. Other symmetry sets of the pair of curves, like the affine area symmetry set and the affine envelope symmetry set can be also used to describe geometric properties of the improper affine sphere.Mathematics Subject Classification (2010). 53A15.

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Envelope of mid-hyperplanes of a hypersurface

Cambraia¹,

Craizer

2017

J. Geom.

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Given 2 points of a smooth hypersurface, their mid-hyperplane is the hyperplane passing through their mid-point and the intersection of their tangent spaces. In this paper we study the envelope of these mid-hyperplanes (EMH) at pairs whose tangent spaces are transversal. We prove that this envelope consists of centers of conics having contact of order at least 3 with the hypersurface at both points. Moreover, we describe general conditions for the EMH to be a smooth hypersurface. These results are extensions of the corresponding well-known results for curves. In the case of curves, if the EMH is contained in a straight line, the curve is necessarily affinely symmetric with respect to the line. We show through a counter-example that this property does not hold for hypersurfaces.Mathematics Subject Classification (2010). 53A15.

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Affinely invariant symmetry sets

Cited by 11 publications

References 16 publications

Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

A geometric representation of improper indefinite affine spheres with singularities

Envelope of mid-hyperplanes of a hypersurface

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