Página 10 dos resultados de 106770 itens digitais encontrados em 0.085 segundos

## Nearly K\"ahler geometry and (2,3,5)-distributions via projective holonomy

Gover, Rod; Panai, Roberto; Willse, Travis
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
We show that any dimension 6 nearly K\"ahler (or nearly para-K\"ahler) geometry arises as a projective manifold equipped with a $\mathrm{G}_2^{(*)}$ holonomy reduction. In the converse direction we show that if a projective manifold is equipped with a parallel 7-dimensional cross product on its standard tractor bundle then the manifold is: a Riemannian nearly K\"ahler manifold, if the cross product is definite; otherwise, if the cross product has the other algebraic type, the manifold is in general stratified with nearly K\"ahler and nearly para-K\"ahler parts separated by a hypersurface which canonically carries a Cartan $(2,3,5)$-distribution. This hypersurface is a projective infinity for the pseudo-Riemannian geometry elsewhere on the manifold, and we establish how the Cartan distribution can be understood explicitly, and also in terms of conformal geometry, as a limit of the ambient nearly (para-)K\"ahler structures. Any real-analytic $(2,3,5)$-distribution is seen to arise as such a limit, because we can solve the geometric Dirichlet problem of building a collar structure equipped with the required holonomy-reduced projective structure. Our approach is to use Cartan/tractor theory to understand all structures as arising from a curved version of the algebra of imaginary (split) octonions as a flat structure over its projectivization. The perspective is used to establish results concerning the projective compactification of nearly (para-)K\"ahler manifolds.; Comment: 57 pages; added missing words at end of Definition 3.1

## Geometry of the restricted Boltzmann machine

Cueto, Maria Angelica; Morton, Jason; Sturmfels, Bernd
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
The restricted Boltzmann machine is a graphical model for binary random variables. Based on a complete bipartite graph separating hidden and observed variables, it is the binary analog to the factor analysis model. We study this graphical model from the perspectives of algebraic statistics and tropical geometry, starting with the observation that its Zariski closure is a Hadamard power of the first secant variety of the Segre variety of projective lines. We derive a dimension formula for the tropicalized model, and we use it to show that the restricted Boltzmann machine is identifiable in many cases. Our methods include coding theory and geometry of linear threshold functions.; Comment: 18 pages, 5 figures, 1 table

## On Klein's So-called Non-Euclidean geometry

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
In two papers titled "On the so-called non-Euclidean geometry", I and II, Felix Klein proposed a construction of the spaces of constant curvature -1, 0 and and 1 (that is, hyperbolic, Euclidean and spherical geometry) within the realm of projective geometry. Klein's work was inspired by ideas of Cayley who derived the distance between two points and the angle between two planes in terms of an arbitrary fixed conic in projective space. We comment on these two papers of Klein and we make relations with other works.; Comment: To appear in : Sophus Lie and Felix Klein: The Erlangen program and its impact in mathematics and physics (ed. L. Ji and A. Papadopoulos), European Mathematical Society Publishing House, 2014

## Infinitesimal Differential Geometry

Giordano, Paolo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
Using standard analysis only, we present an extension ${^\bullet\R}$ of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Differential Geometry, Analysis and Physics. On the other hand we want to show that these infinitesimals may be also useful in infinite dimensional Differential Geometry, e.g. to study spaces of mappings. We define a full embedding of the category Man${}^n$ of finite dimensional $\mathcal{C}^n$ manifolds in a cartesian closed category. In it we have a functor ${}^\bullet (-)$ which extends these spaces adding new infinitesimal points and with values in another full cartesian closed embedding of Man${}^n$. We present a first development of Differential Geometry using these infinitesimals.; Comment: Submitted to: AMUC, December 2003. We added a sheaf property to the definition of $C^n$ space so that now they generalize diffeological spaces and every extended space has now a topology. We also added a final section which compares our construction with other theories of infinitesimals like NSA, SDG and Weil functors

## Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry

Bruasse, Laurent; Teleman, Andrei
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
We give a generalisation of the theory of optimal destabilizing 1-parameter subgroups to non-algebraic complex geometry. Consider a holomorphic action $G\times F\to F$ of a complex reductive Lie group $G$ on a finite dimensional (possibly non-compact) K\"ahler manifold $F$. Using a Hilbert type criterion for the (semi)stability of symplectic actions, we associate to any non semistable point $f\in F$ a unique optimal destabilizing vector in $\g$ and then a naturally defined point $f_0$ which is semistable for the action of a certain reductive subgroup of $G$ on a submanifold of $F$. We get a natural stratification of $F$ which is the analogue of the Shatz stratification for holomorphic vector bundles. In the last chapter we show that our results can be generalized to the gauge theoretical framework: first we show that the system of semistable quotients associated with the classical Harder-Narasimhan filtration of a non-semistable bundle $\EE$ can be recovered as the limit object in the direction given by the optimal destabilizing vector of $\EE$. Second, we extend this principle to holomorphic pairs: we give the analogue of the Harder-Narasimhan theorem for this moduli problem and we discuss the relation between the Harder-Narasimhan filtration of a non-semistable holomorphic pair and its optimal destabilizing vector.; Comment: Latex...

## Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlev\'{e} equation of type VI, Part I

Inaba, Michi-aki; Iwasaki, Katsunori; Saito, Masa-Hiko
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
In this paper, we will give a complete geometric background for the geometry of Painlev\'e $VI$ and Garnier equations. By geometric invariant theory, we will construct a smooth coarse moduli space $M_n^{\balpha}(\bt, \blambda, L)$ of stable parabolic connection on $\BP^1$ with logarithmic poles at $D(\bt) = t_1 + ... + t_n$ as well as its natural compactification. Moreover the moduli space $\cR(\cP_{n, \bt})_{\ba}$ of Jordan equivalence classes of $SL_2(\C)$-representations of the fundamental group $\pi_1(\BP^1 \setminus D(\bt),\ast)$ are defined as the categorical quotient. We define the Riemann-Hilbert correspondence $\RH: M_n^{\balpha}(\bt, \blambda, L) \lra \cR(\cP_{n, \bt})_{\ba}$ and prove that $\RH$ is a bimeromorphic proper surjective analytic map. Painlev\'e and Garnier equations can be derived from the isomonodromic flows and Painlev\'e property of these equations are easily derived from the properties of $\RH$. We also prove that the smooth parts of both moduli spaces have natural symplectic structures and $\RH$ is a symplectic resolution of singularities of $\cR(\cP_{n, \bt})_{\ba}$, from which one can give geometric backgrounds for other interesting phenomena, like Hamiltonian structures, B\"acklund transformations, special solutions of these equations.; Comment: 76 pages...

## Hans Duistermaat's contributions to Poisson geometry

Sjamaar, Reyer
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
Hans Duistermaat was scheduled to lecture in the 2010 School on Poisson Geometry at IMPA, but passed away suddenly. This is a record of a talk I gave at the 2010 Conference on Poisson Geometry (the week after the School) to share some of my memories of him and to give a brief assessment of his impact on the subject.; Comment: 16 pages

## Surfaces in Lie sphere geometry and the stationary Davey-Stewartson hierarchy

Ferapontov, E. V.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
We introduce two basic invariant forms which define generic surface in 3-space uniquely up to Lie sphere equivalence. Two particularly interesting classes of surfaces associated with these invariants are considered, namely, the Lie-minimal surfaces and the diagonally-cyclidic surfaces. For diagonally-cyclidic surfaces we derive the stationary modified Veselov-Novikov equation, whose role in the theory of these surfaces is similar to that of Calapso's equation in the theory of isothermic surfaces. Since Calapso's equation itself turns out to be related to the stationary Davey-Stewartson equation, these results shed some new light on differential geometry of the stationary Davey-Stewartson hierarchy. Diagonally-cyclidic surfaces are the natural Lie sphere analogs of the isothermally-asymptotic surfaces in projective differential geometry for which we also derive the stationary modified Veselov-Novikov equation with the different real reduction. Parallels between invariants of surfaces in Lie sphere geometry and reciprocal invariants of hydrodynamic type systems are drawn in the conclusion.; Comment: Latex, 31 pages

## Geometric Invariant Theory and Birational Geometry

Hu, Yi
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
In this paper we will survey some recent developments in the last decade or so on variation of Geometric Invariant Theory and its applications to Birational Geometry such as the weak Factorization Theorems of nonsingular projective varieties and more generally projective varieties with finite quotient singularities. Along the way, we will also mention some progresses on birational geometry of hyperK\"ahler manifolds as well as certain open problems and conjectures.; Comment: 22 pages

## Reduction of Vaisman structures in complex and quaternionic geometry

Gini, Rosa; Ornea, Liviu; Parton, Maurizio; Piccinni, Paolo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,\Gamma) where K is a Kaehler manifold and \Gamma a discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kaehler manifold (K,\Gamma) as the rank of a natural quotient of \Gamma, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kaehler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover we define locally conformal hyperkaehler reduction as an equivariant version of hyperkaehler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally we show that locally conformal hyperkaehler reduction induces hyperkaehler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperkaehler reduction.; Comment: 29 pages; Section 4 changed (and accordingly the Introduction); Remark 8.2 added; References updated

## The Nature of Length, Area, and Volume in Taxicab Geometry

Thompson, Kevin P.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
While the concept of straight-line length is well understood in taxicab geometry, little research has been done into the length of curves or the nature of area and volume in this geometry. This paper sets forth a comprehensive view of the basic dimensional measures in taxicab geometry.; Comment: 18 pages, 12 figures

## A Note on Distributional Semi-Riemannian Geometry

Steinbauer, Roland
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
We discuss some basic concepts of semi-Riemannian geometry in low-regularity situations. In particular, we compare the settings of (linear) distributional geometry in the sense of L. Schwartz and nonlinear distributional geometry in the sense of J.F. Colombeau.; Comment: 11 pages, Contribution presented at the 12th Serbian Mathematical Congress, Novi Sad, September 2008

## Generalized Kahler Geometry from supersymmetric sigma models

Bredthauer, Andreas; Lindstrom, Ulf; Persson, Jonas; Zabzine, Maxim
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
We give a physical derivation of generalized Kahler geometry. Starting from a supersymmetric nonlinear sigma model, we rederive and explain the results of Gualtieri regarding the equivalence between generalized Kahler geometry and the bi-hermitean geometry of Gates-Hull-Rocek. When cast in the language of supersymmetric sigma models, this relation maps precisely to that between the Lagrangian and the Hamiltonian formalisms. We also discuss topological twist in this context.; Comment: 18 pages

## A metric approach to Fr\'echet geometry

Müller, Olaf
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
The aim of this article is to present the category of bounded Frechet manifolds in respect to which we will review the geometry of Frechet manifolds with a stronger accent on its metric aspect. An inverse function theorem in the sense of Nash and Moser in this category is proved, and some applications to Riemannian geometry are given.; Comment: 30 pages, no figures; submitted to Journal of Geometry and Physics

## Noncommutative Geometry Approach to Principal and Associated Bundles

Baum, Paul F.; Hajac, Piotr M.; Matthes, Rainer; Szymanski, Wojciech
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) C*-algebras. We introduce the concept of piecewise triviality to adapt the standard notion of local triviality to fibre products of C*-algebras. In the context of principal actions, we study in detail an example of a non-proper free action with continuous translation map, and examples of compact principal bundles which are piecewise trivial but not locally trivial, and neither piecewise trivial nor locally trivial, respectively. We show that the module of continuous sections of a vector bundle associated to a compact principal bundle is a cotensor product of the algebra of functions defined on the total space (that are continuous along the base and polynomial along the fibres) with the vector space of the representation. On the algebraic side, we review the formalism of connections for the universal differential algebras. In the differential geometry framework, we consider smooth connections on principal bundles as equivariant splittings of the cotangent bundle, as 1-form-valued derivations of the algebra of smooth functions on the structure group...

## Some relationships between the geometry of the tangent bundle and the geometry of the Riemannian base manifold

Henry, Guillermo; Keilhauer, Guillermo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
We compute the curvature tensor of the tangent bundle of a Riemannian manifold endowed with a natural metric and we get some relationships between the geometry of the base manifold and the geometry of the tangent bundle.; Comment: 15 pages

## D-branes and Azumaya noncommutative geometry: From Polchinski to Grothendieck

Liu, Chien-Hao; Yau, Shing-Tung
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
We review first Azumaya geometry and D-branes in the realm of algebraic geometry along the line of Polchinski-Grothendieck Ansatz from our earlier work and then use it as background to introduce Azumaya $C^{\infty}$-manifolds with a fundamental module and morphisms therefrom to a projective complex manifold. This gives us a description of D-branes of A-type. Donaldson's picture of Lagrangian and special Lagrangian submanifolds as selected from the zero-locus of a moment map on a related space of maps can be merged into the setting. As a pedagogical toy model, we study D-branes of A-type in a Calabi-Yau torus. Simple as it is, it reveals several features of D-branes, including their assembling/disassembling. The 4th theme of Sec. 2.4, the 2nd theme of Sec. 4.2, and Sec. 4.3 are to be read respectively with G\'omez-Sharpe (arXiv:hep-th/0008150), Donagi-Katz-Sharpe (arXiv:hep-th/0309270), and Denef (arXiv:hep-th/0107152). Some string-theoretical remarks are given at the end of each section.; Comment: 58+2 pages, 7 figures

## The geometry of recursion operators

Bande, G.; Kotschick, D.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.09%
We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser's theorem for simultaneous isotopies of two families of symplectic forms. We also consider the geometric structures defined by pairs and triples of symplectic forms for which the squares of the intertwining endomorphisms are plus or minus the identity. For pairs of forms we recover the notions of symplectic pairs and of holomorphic symplectic structures. For triples we recover the notion of a hypersymplectic structure, and we also find three new structures that have not been considered before. One of these is the symplectic formulation of hyper-Kaehler geometry, which turns out to be a strict generalization of the usual definition in terms of differential or Kaehler geometry.; Comment: cosmetic changes only; to appear in Comm. Math. Phys

## Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System

Fassò, Francesco; Giacobbe, Andrea
Tipo: Artigo de Revista Científica