The centre symmetry set

Giblin, Peter; Holtom, P. A.

doi:10.4064/-50-1-91-105

Cited by 22 publications

(31 citation statements)

References 8 publications

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“…The statement on the number of cusps of Wigner caustics was first proved by Berry [3], and the statement on the number of cusps of CSS by Giblin and Holtom [9]. The last inequality of the theorem is new.…”

Section: Theorem 72 Let M Be a Generic Smooth Convex Closed Curve Inmentioning

confidence: 87%

“…where the number of cusps of the CSS and of the Wigner caustic are equal to three and neither curve is self intersecting can be found in [9]. We picture a case when the number of cusps of the Wigner caustic is three and the CSS is self intersecting and the number of its cusps is five, and another case when both the Wigner caustic and the CSS are self intersecting and both have five cusps (Figs.…”

Section: Figures Of Gcs(m)mentioning

confidence: 99%

“…The last inequality of the theorem is new. It follows immediately from the characterization in [9] of cusps of E 1/2 (M) by the curvature ratio being 1 and cusps of CSS of M by the derivative of the curvature ratio being 0, using Rolle's theorem.…”

Section: Theorem 72 Let M Be a Generic Smooth Convex Closed Curve Inmentioning

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%

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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 72 Let M Be a Generic Smooth Convex Closed Curve Inmentioning

confidence: 87%

Section: Figures Of Gcs(m)mentioning

confidence: 99%

Section: Theorem 72 Let M Be a Generic Smooth Convex Closed Curve Inmentioning

confidence: 99%

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

confidence: 99%

See 2 more Smart Citations

Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds

Domitrz

Rios

2013

Geom Dedicata

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“…In [7] we have investigated the central symmetry of X (see also [1], [2], [3]). For p ∈ C 2n we have introduced a number µ(p) of pairs of points x, y ∈ X, such that p is the center of the interval xy.…”

Section: Introductionmentioning

confidence: 99%