Página 10 dos resultados de 106770 itens digitais encontrados em 0.038 segundos

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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## Geometry of the restricted Boltzmann machine

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/08/2009

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/06/2014

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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

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## Infinitesimal Differential Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Mathematics - Complex Variables#Mathematics - Algebraic Geometry#Mathematics - Differential Geometry#Mathematics - Symplectic Geometry#32M05#53D20#14L24#14L30#32L05#32Q15

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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Algebraic Geometry#Mathematics - Differential Geometry#Mathematics - Symplectic Geometry#Nonlinear Sciences - Exactly Solvable and Integrable Systems#34M55, 14D20, 32G34, 34G20, 58F05

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...

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## Hans Duistermaat's contributions to Poisson geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/10/2011

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#Mathematics - History and Overview#Mathematics - Differential Geometry#Mathematics - Symplectic Geometry

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/10/1997

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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

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## Geometric Invariant Theory and Birational Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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## Reduction of Vaisman structures in complex and quaternionic geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/01/2011

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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

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## A Note on Distributional Semi-Riemannian Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/11/2008

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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

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## Generalized Kahler Geometry from supersymmetric sigma models

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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## A metric approach to Fr\'echet geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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## Noncommutative Geometry Approach to Principal and Associated Bundles

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/12/2009

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/03/2010

Relevância na Pesquisa

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#Mathematics - Symplectic Geometry#High Energy Physics - Theory#Mathematics - Algebraic Geometry#53C38, 14A22, 81T30

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

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## The geometry of recursion operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/03/2007

Relevância na Pesquisa

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#Mathematics - Symplectic Geometry#Mathematics - Dynamical Systems#Nonlinear Sciences - Exactly Solvable and Integrable Systems

Bifibrations, in symplectic geometry called also dual pairs, play a relevant
role in the theory of superintegrable Hamiltonian systems. We prove the
existence of an analogous bifibrated geometry in dynamical systems with a
symmetry group such that the reduced dynamics is periodic. The integrability of
such systems has been proven by M. Field and J. Hermans with a reconstruction
technique. We apply the result to the nonholonomic system of a ball rolling on
a surface of revolution.; Comment: This is a contribution to the Proc. of workshop on Geometric Aspects
of Integrable Systems (July 17-19, 2006; Coimbra, Portugal), published in
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA/

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## From real affine geometry to complex geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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We construct from a real affine manifold with singularities (a tropical
manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental
problem in mirror symmetry. Furthermore, a striking feature of our approach is
that it yields an explicit and canonical order-by-order description of the
degeneration via families of tropical trees.
This gives complete control of the B-model side of mirror symmetry in terms
of tropical geometry. For example, we expect our deformation parameter is a
canonical coordinate, and expect period calculations to be expressible in terms
of tropical curves. We anticipate this will lead to a proof of mirror symmetry
via tropical methods.
This paper is the key step of the program we initiated in math.AG/0309070.; Comment: v3: 128 pages, published version

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