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Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds
We study the global centre symmetry set ( GCS ) of a smooth closed submanifold . 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 uses th...
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| Published in: | Geometriae dedicata 2014-04, Vol.169 (1), p.361-382 |
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| Main Authors: | , |
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| container_end_page | 382 |
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| container_title | Geometriae dedicata |
| container_volume | 169 |
| creator | Domitrz, Wojciech Rios, Pedro de M. |
| description | We study the global centre symmetry set (
GCS
) of a smooth closed submanifold
. 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
uses the concept of an affine
-equidistant of
. When
is a Lagrangian submanifold in the affine symplectic space
, we present generating families for singularities of
and prove that the caustic of any simple stable Lagrangian singularity in a
-dimensional Lagrangian fibre bundle is realizable as the germ of an affine equidistant of some
. We characterize the criminant part of
GCS
in terms of bitangent hyperplanes to
. Then, after presenting the appropriate equivalence relation to be used in this Lagrangian case, we classify the affine-Lagrangian stable singularities of
GCS
. In particular we show that, already for a smooth closed convex curve
, many singularities of
GCS
which are affine stable are not affine-Lagrangian stable. |
| doi_str_mv | 10.1007/s10711-013-9861-2 |
| format | article |
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GCS
) of a smooth closed submanifold
. 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
uses the concept of an affine
-equidistant of
. When
is a Lagrangian submanifold in the affine symplectic space
, we present generating families for singularities of
and prove that the caustic of any simple stable Lagrangian singularity in a
-dimensional Lagrangian fibre bundle is realizable as the germ of an affine equidistant of some
. We characterize the criminant part of
GCS
in terms of bitangent hyperplanes to
. Then, after presenting the appropriate equivalence relation to be used in this Lagrangian case, we classify the affine-Lagrangian stable singularities of
GCS
. In particular we show that, already for a smooth closed convex curve
, many singularities of
GCS
which are affine stable are not affine-Lagrangian stable.</description><identifier>ISSN: 0046-5755</identifier><identifier>EISSN: 1572-9168</identifier><identifier>DOI: 10.1007/s10711-013-9861-2</identifier><language>eng</language><publisher>Dordrecht: Springer Netherlands</publisher><subject>Algebraic Geometry ; Convex and Discrete Geometry ; Differential Geometry ; Hyperbolic Geometry ; Mathematics ; Mathematics and Statistics ; Original Paper ; Projective Geometry ; Topology</subject><ispartof>Geometriae dedicata, 2014-04, Vol.169 (1), p.361-382</ispartof><rights>The Author(s) 2013</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c331t-8e5db6dcc1883ab013fb659229f32dfb163b286421390fe42a97137efd3652703</citedby><cites>FETCH-LOGICAL-c331t-8e5db6dcc1883ab013fb659229f32dfb163b286421390fe42a97137efd3652703</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27901,27902</link.rule.ids></links><search><creatorcontrib>Domitrz, Wojciech</creatorcontrib><creatorcontrib>Rios, Pedro de M.</creatorcontrib><title>Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds</title><title>Geometriae dedicata</title><addtitle>Geom Dedicata</addtitle><description>We study the global centre symmetry set (
GCS
) of a smooth closed submanifold
. 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
uses the concept of an affine
-equidistant of
. When
is a Lagrangian submanifold in the affine symplectic space
, we present generating families for singularities of
and prove that the caustic of any simple stable Lagrangian singularity in a
-dimensional Lagrangian fibre bundle is realizable as the germ of an affine equidistant of some
. We characterize the criminant part of
GCS
in terms of bitangent hyperplanes to
. Then, after presenting the appropriate equivalence relation to be used in this Lagrangian case, we classify the affine-Lagrangian stable singularities of
GCS
. In particular we show that, already for a smooth closed convex curve
, many singularities of
GCS
which are affine stable are not affine-Lagrangian stable.</description><subject>Algebraic Geometry</subject><subject>Convex and Discrete Geometry</subject><subject>Differential Geometry</subject><subject>Hyperbolic Geometry</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Original Paper</subject><subject>Projective Geometry</subject><subject>Topology</subject><issn>0046-5755</issn><issn>1572-9168</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><recordid>eNp9kM1OxCAYRYnRxHH0AdzxAigfDNAuzcS_ZBIXauKOQIGGSUsV6GLe3uq4dnU399zcHISugd4Apeq2AFUAhAInbSOBsBO0AqEYaUE2p2hF6UYSoYQ4Rxel7CmlrVJshT5eY-rnweRYoy94Cth_zdHFUk2qBZvkcD9M1gy486lmj8thHH3NB1x8_e3vTJ9N6qNJuMx2NCmGaXDlEp0FMxR_9Zdr9P5w_7Z9IruXx-ft3Y50nEMljRfOStd10DTc2OV_sFK0jLWBMxcsSG5ZIzcMeEuD3zDTKuDKB8elYIryNYLjbpenUrIP-jPH0eSDBqp_1OijGr1M6x81mi0MOzJl6abeZ72f5pyWm_9A31k9Z_A</recordid><startdate>20140401</startdate><enddate>20140401</enddate><creator>Domitrz, Wojciech</creator><creator>Rios, Pedro de M.</creator><general>Springer Netherlands</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20140401</creationdate><title>Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds</title><author>Domitrz, Wojciech ; Rios, Pedro de M.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c331t-8e5db6dcc1883ab013fb659229f32dfb163b286421390fe42a97137efd3652703</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2014</creationdate><topic>Algebraic Geometry</topic><topic>Convex and Discrete Geometry</topic><topic>Differential Geometry</topic><topic>Hyperbolic Geometry</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Paper</topic><topic>Projective Geometry</topic><topic>Topology</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Domitrz, Wojciech</creatorcontrib><creatorcontrib>Rios, Pedro de M.</creatorcontrib><collection>SpringerOpen</collection><collection>CrossRef</collection><jtitle>Geometriae dedicata</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Domitrz, Wojciech</au><au>Rios, Pedro de M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds</atitle><jtitle>Geometriae dedicata</jtitle><stitle>Geom Dedicata</stitle><date>2014-04-01</date><risdate>2014</risdate><volume>169</volume><issue>1</issue><spage>361</spage><epage>382</epage><pages>361-382</pages><issn>0046-5755</issn><eissn>1572-9168</eissn><abstract>We study the global centre symmetry set (
GCS
) of a smooth closed submanifold
. 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
uses the concept of an affine
-equidistant of
. When
is a Lagrangian submanifold in the affine symplectic space
, we present generating families for singularities of
and prove that the caustic of any simple stable Lagrangian singularity in a
-dimensional Lagrangian fibre bundle is realizable as the germ of an affine equidistant of some
. We characterize the criminant part of
GCS
in terms of bitangent hyperplanes to
. Then, after presenting the appropriate equivalence relation to be used in this Lagrangian case, we classify the affine-Lagrangian stable singularities of
GCS
. In particular we show that, already for a smooth closed convex curve
, many singularities of
GCS
which are affine stable are not affine-Lagrangian stable.</abstract><cop>Dordrecht</cop><pub>Springer Netherlands</pub><doi>10.1007/s10711-013-9861-2</doi><tpages>22</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | Geometriae dedicata, 2014-04, Vol.169 (1), p.361-382 |
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| source | Springer Nature |
| subjects | Algebraic Geometry Convex and Discrete Geometry Differential Geometry Hyperbolic Geometry Mathematics Mathematics and Statistics Original Paper Projective Geometry Topology |
| title | Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds |
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