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Special Kaehler geometry

Van Proeyen, Antoine
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 15/02/2000
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The geometry that is defined by the scalars in couplings of Einstein-Maxwell theories in N=2 supergravity in 4 dimensions is denoted as special Kaehler geometry. There are several equivalent definitions, the most elegant ones involve the symplectic duality group. The original construction used conformal symmetry, which immediately clarifies the symplectic structure and provides a way to make connections to quaternionic geometry and Sasakian manifolds.; Comment: Contribution to the Proceedings of the Second Meeting on Quaternionic Structures in Mathematics and Physics, Rome 6-10 September 1999; 17 pages, 1 figure, using color (colordvi.sty)

Geometry of Distributions and $F-$Gordon equation

Nadjafikhah, Mehdi; Aghayan, Reza
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 25/08/2009
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In this paper we describe the geometry of distributions by their symmetries, and present a simplified proof of the Frobenius theorem and some related corollaries. Then, we study the geometry of solutions of $F-$Gordon equation; A PDE which appears in differential geometry and relativistic field theory.; Comment: 28 pages

Geometry of Maslov cycles

Barilari, Davide; Lerario, Antonio
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 02/01/2013
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We introduce the notion of induced Maslov cycle, which describes and unifies geometrical and topological invariants of many apparently unrelated problems, from Real Algebraic Geometry to sub-Riemannian Geometry.

Exceptional Calabi--Yau spaces: the geometry of $\mathcal{N}=2$ backgrounds with flux

Ashmore, Anthony; Waldram, Daniel
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 30/09/2015
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36.09%
In this paper we define the analogue of Calabi--Yau geometry for generic $D=4$, $\mathcal{N}=2$ flux backgrounds in type II supergravity and M-theory. We show that solutions of the Killing spinor equations are in one-to-one correspondence with integrable, globally defined structures in $E_{7(7)}\times\mathbb{R}^+$ generalised geometry. Such "exceptional Calabi--Yau" geometries are determined by two generalised objects that parametrise hyper- and vector-multiplet degrees of freedom and generalise conventional complex, symplectic and hyper-Kahler geometries. The integrability conditions for both hyper- and vector-multiplet structures are given by the vanishing of moment maps for the "generalised diffeomorphism group" of diffeomorphisms combined with gauge transformations. We give a number of explicit examples and discuss the structure of the moduli spaces of solutions. We then extend our construction to $D=5$ and $D=6$ flux backgrounds preserving eight supercharges, where similar structures appear, and finally discuss the analogous structures in $O(d,d)\times\mathbb{R}^+$ generalised geometry.; Comment: 68 pages plus appendices

Algorithms in Real Algebraic Geometry: A Survey

Basu, Saugata
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 04/09/2014
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36.09%
We survey both old and new developments in the theory of algorithms in real algebraic geometry -- starting from effective quantifier elimination in the first order theory of reals due to Tarski and Seidenberg, to more recent algorithms for computing topological invariants of semi-algebraic sets. We emphasize throughout the complexity aspects of these algorithms and also discuss the computational hardness of the underlying problems. We also describe some recent results linking the computational hardness of decision problems in the first order theory of the reals, with that of computing certain topological invariants of semi-algebraic sets. Even though we mostly concentrate on exact algorithms, we also discuss some numerical approaches involving semi-definite programming that have gained popularity in recent times.; Comment: 41 pages, 4 figures. Based on survey talk given at the Real Algebraic Geometry Conference, Rennes, June 20-24, 2011. Some references updated and some newer material added

Key developments in geometry in the 19th Century

Wells Jr, Raymond O.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 03/01/2013
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This paper describes several key discoveries in the 19th century that led to the modern theory of manifolds in the twentieth century: intrinsic differential geometry, projective geometry and higher dimensional manifolds and Riemannian geometry.

Lectures on Arithmetic Noncommutative Geometry

Marcolli, Matilde
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 27/09/2004
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This is the text of a series of five lectures given by the author at the "Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras" held at Vanderbilt University in May 2004. It is meant as an overview of recent results illustrating the interplay between noncommutative geometry and arithmetic geometry/number theory.; Comment: 129 pages LaTeX, 28 figures

An XML-Format for Conjectures in Geometry (Work-in-Progress)

Quaresma, Pedro
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 10/07/2012
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With a large number of software tools dedicated to the visualisation and/or demonstration of properties of geometric constructions and also with the emerging of repositories of geometric constructions, there is a strong need of linking them, and making them and their corpora, widely usable. A common setting for interoperable interactive geometry was already proposed, the i2g format, but, in this format, the conjectures and proofs counterparts are missing. A common format capable of linking all the tools in the field of geometry is missing. In this paper an extension of the i2g format is proposed, this extension is capable of describing not only the geometric constructions but also the geometric conjectures. The integration of this format into the Web-based GeoThms, TGTP and Web Geometry Laboratory systems is also discussed.; Comment: Conferences on Intelligent Computer Mathematics, CICM 2012, 8.-13. July 2012, Jacobs University, Bremen, Germany. 12 pages 2 figures

On the works of Euler and his followers on spherical geometry

Papadopoulos, Athanase
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/09/2014
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We review and comment on some works of Euler and his followers on spherical geometry. We start by presenting some memoirs of Euler on spherical trigonometry. We comment on Euler's use of the methods of the calculus of variations in spherical trigonometry. We then survey a series of geometrical resuls, where the stress is on the analogy between the results in spherical geometry and the corresponding results in Euclidean geometry. We elaborate on two such results. The first one, known as Lexell's Theorem (Lexell was a student of Euler), concerns the locus of the vertices of a spherical triangle with a fixed area and a given base. This is the spherical counterpart of a result in Euclid's Elements, but it is much more difficult to prove than its Euclidean analogue. The second result, due to Euler, is the spherical analogue of a generalization of a theorem of Pappus (Proposition 117 of Book VII of the Collection) on the construction of a triangle inscribed in a circle whose sides are contained in three lines that pass through three given points. Both results have many ramifications, involving several mathematicians, and we mention some of these developments. We also comment on three papers of Euler on projections of the sphere on the Euclidean plane that are related with the art of drawing geographical maps.; Comment: To appear in Ganita Bharati (Indian Mathematics)...

String geometry vs. spin geometry on loop spaces

Waldorf, Konrad
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We introduce various versions of spin structures on free loop spaces of smooth manifolds, based on a classical notion due to Killingback, and additionally coupled to two relations between loops: thin homotopies and loop fusion. The central result of this article is an equivalence between these enhanced versions of spin structures on the loop space and string structures on the manifold itself. The equivalence exists in two settings: in a purely topological one and a in geometrical one that includes spin connections and string connections. Our results provide a consistent, functorial, one-to-one dictionary between string geometry and spin geometry on loop spaces.; Comment: 54 pages. In v2 two errorneous lemmata (2.3.3 and 3.1.3) have been removed, with corresponding changes in Prop. 2.3.4 and Def. 3.1.4 (now Prop. 2.3.3 and Def. 3.1.3, respectively); otherwise minor changes. v3 comes with few minor changes and is the published version

Radar orthogonality and radar length in Finsler and metric spacetime geometry

Pfeifer, Christian
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
The radar experiment connects the geometry of spacetime with an observers measurement of spatial length. We investigate the radar experiment on Finsler spacetimes which leads to a general definition of radar orthogonality and radar length. The directions radar orthogonal to an observer form the spatial equal time surface an observer experiences and the radar length is the physical length the observer associates to spatial objects. We demonstrate these concepts on a forth order polynomial Finsler spacetime geometry which may emerge from area metric or pre-metric linear electrodynamics or in quantum gravity phenomenology. In an explicit generalisation of Minkowski spacetime geometry we derive the deviation from the euclidean spatial length measure in an observers rest frame explicitly.; Comment: 18 pages, 7 figures, axes label in figures corrected, journal references added

Geometry of contact transformations and domains: orderability versus squeezing

Eliashberg, Yakov; Kim, Sang Seon; Polterovich, Leonid
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
Gromov's famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this result in contact geometry? Our main finding is that the answer depends on the sizes of the domains in question: We establish contact non-squeezing on large scales, and show that it disappears on small scales. The algebraic counterpart of the (non)-squeezing problem for contact domains is the question of existence of a natural partial order on the universal cover of the contactomorphisms group of a contact manifold. In contrast to our earlier beliefs, we show that the answer to this question is very sensitive to the topology of the manifold. For instance, we prove that the standard contact sphere is non-orderable while the real projective space is known to be orderable. Our methods include a new embedding technique in contact geometry as well as a generalized Floer homology theory which contains both cylindrical contact homology and Hamiltonian Floer homology. We discuss links to a number of miscellaneous topics such as topology of free loops spaces, quantum mechanics and semigroups. An erratum is attached whose purpose is to is to correct a number of inconsistencies in the main paper. These are related to the grading of generalized Floer homology and do not affect formulations and proofs of the main results of the paper.; Comment: This is the version published by Geometry & Topology on 28 October 2006 and includes the erratum published 1 February 2009

When symplectic topology meets Banach space geometry

Ostrover, Yaron
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 28/04/2014
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36.09%
In this paper we survey some recent works that take the first steps toward establishing bilateral connections between symplectic geometry and several other fields, namely, asymptotic geometric analysis, classical convex geometry, and the theory of normed spaces.; Comment: Submitted to Proceedings of the ICM 2014, 23 pages

On geometry of curves of flags of constant type

Doubrov, Boris; Zelenko, Igor
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 02/10/2011
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36.09%
We develop an algebraic version of Cartan method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space $W$ with respect to the action of a subgroup $G$ of the $GL(W)$. Under some natural assumptions on the subgroup $G$ and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure Linear Algebra. The scope of applicability of the theory includes geometry of natural classes of curves of flags with respect to reductive linear groups or their parabolic subgroups. As simplest examples, this includes the projective and affine geometry of curves. The case of classical groups is considered in more detail.; Comment: 39 pages

On different curvatures of spheres in Funk geometry

Olin, Eugeny A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
We compute the series expansions for the normal curvatures of hyperspheres, the Finsler and Rund curvatures of circles in Funk geometry as the radii tend to infinity. These three curvatures are different at infinity in Funk geometry.; Comment: 15 pages, revised version - references have been corrected. arXiv admin note: substantial text overlap with arXiv:1105.1865

Geometry of manifolds with area metric: multi-metric backgrounds

Schuller, Frederic P.; Wohlfarth, Mattias N. R.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
We construct the differential geometry of smooth manifolds equipped with an algebraic curvature map acting as an area measure. Area metric geometry provides a spacetime structure suitable for the discussion of gauge theories and strings, and is considerably more general than Lorentzian geometry. Our construction of geometrically relevant objects, such as an area metric compatible connection and derived tensors, makes essential use of a decomposition theorem due to Gilkey, whereby we generate the area metric from a finite collection of metrics. Employing curvature invariants for multi-metric backgrounds we devise a class of gravity theories with inherently stringy character, and discuss gauge matter actions.; Comment: 34 pages, REVTeX4, journal version

Fractional Almost Kahler - Lagrange Geometry

Baleanu, Dumitru; Vacaru, Sergiu I.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
The goal of this paper is to encode equivalently the fractional Lagrange dynamics as a nonholonomic almost Kahler geometry. We use the fractional Caputo derivative generalized for nontrivial nonlinear connections (N-connections) originally introduced in Finsler geometry, with further developments in Lagrange and Hamilton geometry and, in our approach, with fractional derivatives. For fundamental geometric objects induced canonically by regular Lagrange functions, we construct compatible almost symplectic forms and linear connections completely determined by a "prime" Lagrange (in particular, Finsler) generating function. We emphasize the importance of such constructions for deformation quantization of fractional Lagrange geometries and applications in modern physics.; Comment: latex2e, 17 pages, v3 performed following requests of referee with additional references and explanations; accepted to "Nonlinear Dynamics"

Birational geometry for number theorists

Abramovich, Dan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Awfully idiosyncratic lecture notes from CMI summer school in arithmetic geometry July 31-August 4, 2006. Does not include: rationality problems, techniques of the minimal model problem and much of the rest. Includes: Lecture 0: geometry and arithmetic of curves Lecture 1: Kodaira dimension and properties, rational connectendess, Lang's and Campana's conjectures. Lecture 2: Campana's program; Campana constellations framed in terms of b-divisors, to allow for a definition of Kodaira dimension directly on the base. A speculative notion of firmaments which may deserve further investigation, especially the arithmetic side. Lecture 3: the minimal model program: very short discussion of bend-and-break; even shorter discussion of finite generation and the existence of flip. Lecture 4: Vojta's conjectures, Campana's conjectures, and ABC.; Comment: CMI summer school in arithmetic geometry, Gottingen, 2006. Numerous corrections following referee's report

One dimensional metrical geometry

Wildberger, N J
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 11/01/2007
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36.09%
One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this situation, such as the Triple quad formula, the Triple spread formula and the Spread polynomials, which are universal analogs of the Chebyshev polynomials of the first kind. Chromogeometry appears here, and the related metrical and algebraic properties of the projective line are brought to the fore.; Comment: 19 pages

Group-like objects in Poisson geometry and algebra

Blohmann, Christian; Weinstein, Alan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 18/01/2007
Relevância na Pesquisa
36.09%
A group, defined as set with associative multiplication and inverse, is a natural structure describing the symmetry of a space. The concept of group generalizes to group objects internal to other categories than sets. But there are yet more general objects that can still be thought of as groups in many ways, such as quantum groups. We explain some of the generalizations of groups which arise in Poisson geometry and quantization: the germ of a topological group, Poisson Lie groups, rigid monoidal structures on symplectic realizations, groupoids, 2-groups, stacky Lie groups, and hopfish algebras.; Comment: 21 pages, based on lectures at School on Poisson Geometry and Related Topics, Keio University, 2006