An Infinite Family of Compact, Complete, and Locally Affine <i>k</i>-Symplectic Manifolds of Dimension Three

We study the complete, compact, locally affine manifolds equipped with a <i>k</i>-symplectic structure, which are the quotients of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvar...

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Autores principales:	Fanich El Mokhtar, Essabab Said
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	<i>k</i>-symplectic structure locally affine manifolds foliations Lagrangian submanifolds Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/4726aececeea4d4ea9b4d52ea2d6fc87
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spelling	oai:doaj.org-article:4726aececeea4d4ea9b4d52ea2d6fc872021-11-25T19:07:13ZAn Infinite Family of Compact, Complete, and Locally Affine <i>k</i>-Symplectic Manifolds of Dimension Three10.3390/sym131121592073-8994https://doaj.org/article/4726aececeea4d4ea9b4d52ea2d6fc872021-11-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2159https://doaj.org/toc/2073-8994We study the complete, compact, locally affine manifolds equipped with a <i>k</i>-symplectic structure, which are the quotients of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msup></semantics></math></inline-formula> by a subgroup <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Γ</mi></semantics></math></inline-formula> of the affine group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>(</mo><mi>n</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msup></semantics></math></inline-formula> acting freely and properly discontinuously on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msup></semantics></math></inline-formula> and leaving invariant the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="italic">k</mi></semantics></math></inline-formula>-symplectic structure, then we construct and give some examples and properties of compact, complete, locally affine two-symplectic manifolds of dimension three.Fanich El MokhtarEssabab SaidMDPI AGarticle<i>k</i>-symplectic structurelocally affine manifoldsfoliationsLagrangian submanifoldsMathematicsQA1-939ENSymmetry, Vol 13, Iss 2159, p 2159 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	<i>k</i>-symplectic structure locally affine manifolds foliations Lagrangian submanifolds Mathematics QA1-939
spellingShingle	<i>k</i>-symplectic structure locally affine manifolds foliations Lagrangian submanifolds Mathematics QA1-939 Fanich El Mokhtar Essabab Said An Infinite Family of Compact, Complete, and Locally Affine <i>k</i>-Symplectic Manifolds of Dimension Three
description	We study the complete, compact, locally affine manifolds equipped with a <i>k</i>-symplectic structure, which are the quotients of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msup></semantics></math></inline-formula> by a subgroup <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Γ</mi></semantics></math></inline-formula> of the affine group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>(</mo><mi>n</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msup></semantics></math></inline-formula> acting freely and properly discontinuously on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mi>n</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msup></semantics></math></inline-formula> and leaving invariant the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="italic">k</mi></semantics></math></inline-formula>-symplectic structure, then we construct and give some examples and properties of compact, complete, locally affine two-symplectic manifolds of dimension three.
format	article
author	Fanich El Mokhtar Essabab Said
author_facet	Fanich El Mokhtar Essabab Said
author_sort	Fanich El Mokhtar
title	An Infinite Family of Compact, Complete, and Locally Affine <i>k</i>-Symplectic Manifolds of Dimension Three
title_short	An Infinite Family of Compact, Complete, and Locally Affine <i>k</i>-Symplectic Manifolds of Dimension Three
title_full	An Infinite Family of Compact, Complete, and Locally Affine <i>k</i>-Symplectic Manifolds of Dimension Three
title_fullStr	An Infinite Family of Compact, Complete, and Locally Affine <i>k</i>-Symplectic Manifolds of Dimension Three
title_full_unstemmed	An Infinite Family of Compact, Complete, and Locally Affine <i>k</i>-Symplectic Manifolds of Dimension Three
title_sort	infinite family of compact, complete, and locally affine <i>k</i>-symplectic manifolds of dimension three
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/4726aececeea4d4ea9b4d52ea2d6fc87
work_keys_str_mv	AT fanichelmokhtar aninfinitefamilyofcompactcompleteandlocallyaffineikisymplecticmanifoldsofdimensionthree AT essababsaid aninfinitefamilyofcompactcompleteandlocallyaffineikisymplecticmanifoldsofdimensionthree AT fanichelmokhtar infinitefamilyofcompactcompleteandlocallyaffineikisymplecticmanifoldsofdimensionthree AT essababsaid infinitefamilyofcompactcompleteandlocallyaffineikisymplecticmanifoldsofdimensionthree
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An Infinite Family of Compact, Complete, and Locally Affine <i>k</i>-Symplectic Manifolds of Dimension Three

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