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Reeb Vector Field
In mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution ... including: * in a contact manifold, given a contact 1-form \alpha, the Reeb vector field satisfies R \in \mathrm\ d\alpha, \ \alpha (R) = 1 ,http://www2.im.uj.edu.pl/katedry/K.G/AutumnSchool/Monday.pdf * in particular, in the context of Sasakian manifold#The Reeb vector field. References * * Contact geometry {{differential-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georges Reeb
Georges Henri Reeb (12 November 1920 – 6 November 1993) was a French mathematician. He worked in differential topology, differential geometry, differential equations, topological dynamical systems theory and non-standard analysis. Biography Reeb was born in Saverne, Bas-Rhin, Alsace, to Theobald Reeb and Caroline Engel. He started studying mathematics at University of Strasbourg, but in 1939 the entire university was evacuated to Clermont-Ferrand due to the German occupation of France. After the war, he completed his studies and in 1948 he defended his PhD thesis, entitled ''Propriétés topologiques des variétés feuilletées'' opological properties of foliated manifoldsand supervised by Charles Ehresmann. In 1952 Reeb was appointed professor at Université Joseph Fourier in Grenoble and in 1954 he visited the Institute for Advanced Study. From 1963 he worked at Université Louis Pasteur in Strasbourg. There, in 1965 he created with Jean Leray and Pierre Lelong the ser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contact Geometry
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution (differential geometry), distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'integrable system, complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem (differential topology), Frobenius theorem. Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contact Manifold
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for ' complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem. Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sasakian Manifold
In differential geometry, a Sasakian manifold (named after Shigeo Sasaki) is a contact manifold (M,\theta) equipped with a special kind of Riemannian metric g, called a ''Sasakian'' metric. Definition A Sasakian metric is defined using the construction of the ''Riemannian cone''. Given a Riemannian manifold (M,g), its Riemannian cone is the product :(M\times ^)\, of M with a half-line ^, equipped with the ''cone metric'' : t^2 g + dt^2,\, where t is the parameter in ^. A manifold M equipped with a 1-form \theta is contact if and only if the 2-form :t^2\,d\theta + 2t\, dt \cdot \theta\, on its cone is symplectic (this is one of the possible definitions of a contact structure). A contact Riemannian manifold is Sasakian, if its Riemannian cone with the cone metric is a Kähler manifold with Kähler form :t^2\,d\theta + 2t\,dt \cdot \theta. Examples As an example consider :S^\hookrightarrow ^=^ where the right hand side is a natural Kähler manifold and read as the cone ove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhäuser
Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (particularly: history of science, geosciences, computer science) and mathematics books and journals under the Birkhäuser imprint (with a leaf logo) sometimes called Birkhäuser Science. * Birkhäuser Verlag – an architecture and design publishing company was (re)created in 2010 when Springer sold its design and architecture segment to ACTAR. The resulting Spanish-Swiss company was then called ActarBirkhäuser. After a bankruptcy, in 2012 Birkhäuser Verlag was sold again, this time to De Gruyter. Additionally, the Reinach-based printer Birkhäuser+GBC operates independently of the above, being now owned by ''Basler Zeitung''. History The original Swiss publishers program focused on regional literature. In the 1920s the sons of Emil Birkhà ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books by decree in 1586, it is the second oldest university press after Cambridge University Press. It is a department of the University of Oxford and is governed by a group of 15 academics known as the Delegates of the Press, who are appointed by the vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, opposite Somerville College, in the inner suburb of Jericho. For the last 500 years, OUP has primarily focused on the publication of pedagogical texts and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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