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Sobre a Geometria de Imersões RiemannianasSantos, Fábio Reis dos Santos 26 May 2015 (has links)
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Previous issue date: 2015-05-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Our purpose is to study the geometry of Riemannian immersions in certain semi-
Riemannian manifolds. Initially, considering linearWeingarten hypersurfaces immersed
in locally symmetric manifolds and, imposing suitable constraints on the scalar curvature,
we guarantee that such a hypersurface is either totally umbilical or isometric to
a isoparametric hypersurface with two distinct principal curvatures, one of them being
simple. In higher codimension, we use a Simons type formula to obtain new characterizations
of hyperbolic cylinders through the study of submanifolds having parallel
normalized mean curvature vector field in a semi-Riemannian space form. Finally,
we investigate the rigidity of complete spacelike hypersurfaces immersed in the steady
state space via applications of some maximum principles. / Nos propomos estudar a geometria de imersões Riemannianas em certas variedades
semi-Riemannianas. Inicialmente, consideramos hipersuperfícies Weingarten
lineares imersas em variedades localmente simétricas e, impondo restrições apropriadas
à curvatura escalar, garantimos que uma tal hipersuperfície é totalmente umbílica
ou isométrica a uma hipersuperfície isoparamétrica com duas curvaturas principais distintas,
sendo uma destas simples. Em codimensão alta, usamos uma fórmula do tipo
Simons para obter novas caracterizações de cilindros hiperbólicos a partir do estudo de
subvariedades com vetor curvatura média normalizado paralelo em uma forma espacial
semi-Riemanniana. Finalmente, investigamos a rigidez de hipersuperfícies tipo-espaço
completas imersas no steady state space via aplicações de alguns princípios do máximo.
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Contributions à l'étude des sous-variétés aléatoires / Contributions to the study of random submanifoldsLetendre, Thomas 24 November 2016 (has links)
Dans cette thèse, nous étudions le volume et la caractéristique d'Euler de sous-variétés aléatoires de codimension r ∈ {1, . . . , n} dans une variété ambiante M de dimension n. Dans un premier modèle, dit des ondes riemanniennes aléatoires, M est une variété riemannienne fermée. Nous considérons alors le lieu Zλ des zéros communs de r combinaisons linéaires aléatoires indépendantes de fonctions propres du laplacien associées à des valeurs propres inférieures à λ 0. Nous obtenons alors les asymptotiques du volume moyen et de la caractéristique d'Euler moyenne de Zλ lorsque λ tend vers l'infini. Dans un second modèle, M est le lieu réel d'une variété projective définie sur les réels. On s'intéresse dans ce cadre au lieu d'annulation réel Zd d'une section holomorphe réelle globale aléatoire de E⊗Ld, où E est un fibré hermitien de rang r, L est un fibré en droites hermitien ample et tous deux sont définis sur les réels. Nous estimons alors les moyennes du volume et de la caractéristique d'Euler de Zd quand d tend vers l'infini. Dans ce modèle algébrique réel, nous calculons aussi l'asymptotique de la variance du volume de Zd pour 1 r < n. Nous en déduisons, dans ce cas, des résultats asymptotiques d'équidistribution de Zd dans M / We study the volume and Euler characteristic of codimension r ∈ {1, . . . , n} random submanifolds in a dimension n manifold M. First, we consider Riemannian random waves. That is M is a closed Riemannian manifold and we study the common zero set Zλ of r independent random linear combinations of eigenfunctions of the Laplacian associated to eigenvalues smaller than λ 0. We compute estimates for the mean volume and Euler characteristic of Zλ as λ goes to infinity. We also consider a model of random real algebraic manifolds. In this setting, M is the real locus of a projective manifold defined over the reals. Then, we consider the real vanishing locus Zd of a random real global holomorphic section of E ⊗ Ld, where E is a rank r Hermitian vector bundle, L is an ample Hermitian line bundle and both these bundles are defined over the reals. We compute the asymptotics of the mean volume and Euler characteristic of Zd as d goes to infinity. In this real algebraic setting, we also compute the asymptotic of the variance of the volume of Zd, when 1 r < n. In this case, we prove asympotic equidistribution results for Zd in M
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Abelianization and Floer homology of Lagrangians in clean intersectionSchmäschke, Felix 14 December 2016 (has links)
This thesis is split up into two parts each revolving around Floer
homology and quantum cohomology of closed monotone symplectic
manifolds. In the first part we consider symplectic manifolds obtained
by symplectic reduction. Our main result is that a quantum version of
an abelianization formula of Martin holds, which relates
the quantum cohomologies of symplectic quotients by a group and by its
maximal torus. Also we show a quantum version of the Leray-Hirsch
theorem for Floer homology of Lagrangian intersections in the
quotient.
The second part is devoted to Floer homology of a pair of monotone
Lagrangian submanifolds in clean intersection. Under these assumptions
the symplectic action functional is degenerated. Nevertheless
Frauenfelder defines a version of Floer
homology, which is in a certain sense an infinite dimensional analogon
of Morse-Bott homology. Via natural filtrations on the chain level we
were able to define two spectral sequences which serve as a tool to
compute Floer homology. We show how these are used to obtain new
intersection results for simply connected Lagrangians in the product
of two complex projective spaces.
The link between both parts is that in the background the same
technical methods are applied; namely the theory of holomorphic strips
with boundary on Lagrangians in clean intersection. Since all our
constructions rely heavily on these methods we also give a detailed
account of this theory although in principle many results are not new
or require only straight forward generalizations.:1. Introduction
2. Overview of the main results
2.1. Abelianization .
2.2. Quantum Leray-Hirsch theorem
2.3. Floer homology of Lagrangians in clean intersection
3. Background
3.1. Symplectic geometry .
3.2. Hamiltonian action functional
3.3. Morse homology .
3.4. Floer homology
4. Asymptotic analysis
4.1. Main statement .
4.2. Mean-value inequality .
4.3. Isoperimetric inequality
4.4. Linear theory
4.5. Proofs
5. Compactness
5.1. Cauchy-Riemann-Floer equation .
5.2. Local convergence .
5.3. Convergence on the ends
5.4. Minimal energy .
5.5. Action, energy and index estimates
6. Fredholm Theory
6.1. Banach manifold .
6.2. Linear theory
7. Transversality
7.1. Setup
7.2. R-dependent structures
7.3. R-invariant structures .
7.4. Regular points .
7.5. Floer’s ε-norm .
8. Gluing
8.1. Setup and main statement
8.2. Pregluing .
8.3. A uniform bounded right inverse
8.4. Quadratic estimate
8.5. Continuity of the gluing map
8.6. Surjectivity of the gluing map
8.7. Degree of the gluing map
8.8. Morse gluing .
9. Orientations
9.1. Preliminaries and notation
9.2. Spin structures and relative spin structures
9.3. Orientation of caps
9.4. Linear theory .
10.Pearl homology
10.1. Overview .
10.2. Pearl trajectories .
10.3. Invariance .
10.4. Spectral sequences
11.Proofs of the main results
11.1. Abelianization Theorem
11.2. Quantum Leray-Hirsch Theorem .
12.Applications
12.1. Quantum cohomology of the complex Grassmannian
12.2. Lagrangian spheres in symplectic quotients
A. Estimates
A.1. Derivative of the exponential map
A.2. Parallel Transport
A.3. Estimates for strips
B. Operators on Hilbert spaces
B.1. Spectral gap
B.2. Flow operator
C. Viterbo index
D. Quotients of principal bundles by maximal tori
D.1. Compact Lie groups
D.2. The cohomology of the quotient of principle bundles by maximal tori
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The differential geometry of the fibres of an almost contract metric submersionTshikunguila, Tshikuna-Matamba 10 1900 (has links)
Almost contact metric submersions constitute a class of Riemannian submersions whose
total space is an almost contact metric manifold. Regarding the base space, two types
are studied. Submersions of type I are those whose base space is an almost contact
metric manifold while, when the base space is an almost Hermitian manifold, then the
submersion is said to be of type II.
After recalling the known notions and fundamental properties to be used in the
sequel, relationships between the structure of the fibres with that of the total space
are established. When the fibres are almost Hermitian manifolds, which occur in the
case of a type I submersions, we determine the classes of submersions whose fibres
are Kählerian, almost Kählerian, nearly Kählerian, quasi Kählerian, locally conformal
(almost) Kählerian, Gi-manifolds and so on. This can be viewed as a classification of
submersions of type I based upon the structure of the fibres.
Concerning the fibres of a type II submersions, which are almost contact metric
manifolds, we discuss how they inherit the structure of the total space.
Considering the curvature property on the total space, we determine its corresponding
on the fibres in the case of a type I submersions. For instance, the cosymplectic
curvature property on the total space corresponds to the Kähler identity on the fibres.
Similar results are obtained for Sasakian and Kenmotsu curvature properties.
After producing the classes of submersions with minimal, superminimal or umbilical
fibres, their impacts on the total or the base space are established. The minimality of
the fibres facilitates the transference of the structure from the total to the base space.
Similarly, the superminimality of the fibres facilitates the transference of the structure
from the base to the total space. Also, it is shown to be a way to study the integrability
of the horizontal distribution.
Totally contact umbilicity of the fibres leads to the asymptotic directions on the total
space.
Submersions of contact CR-submanifolds of quasi-K-cosymplectic and
quasi-Kenmotsu manifolds are studied. Certain distributions of the under consideration
submersions induce the CR-product on the total space. / Mathematical Sciences / D. Phil. (Mathematics)
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Source spaces and perturbations for cluster complexesCharest, François 11 1900 (has links)
Dans ce travail, nous définissons des objets composés de disques complexes
marqués reliés entre eux par des segments de droite munis d’une longueur.
Nous construisons deux séries d’espaces de module de ces objets appelés clus-
ters, une qui sera dite non symétrique, la version ⊗, et l’autre qui est dite
symétrique, la version •. Cette construction permet des choix de perturba-
tions pour deux versions correspondantes des trajectoires de Floer introduites
par Cornea et Lalonde ([CL]). Ces choix devraient fournir une nouvelle option
pour la description géométrique des structures A∞ et L∞ obstruées étudiées
par Fukaya, Oh, Ohta et Ono ([FOOO2],[FOOO]) et Cho ([Cho]).
Dans le cas où L ⊂ (M, ω) est une sous-variété lagrangienne Pin± mono-
tone avec nombre de Maslov ≥ 2, nous définissons une structure d’algèbre A∞
sur les points critiques d’une fonction de Morse générique sur L. Cette struc-
ture est présentée comme une extension du complexe des perles de Oh ([Oh])
muni de son produit quantique, plus récemment étudié par Biran et Cornea
([BC]). Plus généralement, nous décrivons une version géométrique d’une
catégorie de Fukaya avec seul objet L qui se veut alternative à la description
(relative) hamiltonienne de Seidel ([Sei]). Nous vérifions la fonctorialité de
notre construction en définissant des espaces de module de clusters occultés
qui servent d’espaces sources pour des morphismes de comparaison. / We define objects made of marked complex disks connected by metric line seg-
ments and construct two sequences of moduli spaces of these objects, referred
as the ⊗ version (nonsymmetric) and the • version (symmetric). This allows
choices of coherent perturbations over the corresponding versions of the Floer
trajectories proposed by Cornea and Lalonde ([CL]). These perturbations are
intended to lead to an alternative geometric description of the (obstructed) A∞
and L∞ structures studied by Fukaya, Oh, Ohta and Ono ([FOOO2],[FOOO])
and Cho ([Cho]).
Given a Pin± monotone lagrangian submanifold L ⊂ (M, ω) with mini-
mal Maslov number ≥ 2, we define an A∞ -algebra structure from the critical
points of a generic Morse function on L. We express this structure as a cochain
complex extending the pearl complex introduced by Oh ([Oh]) and further ex-
plicited by Biran and Cornea ([BC]), equipped with its quantum product. This
could also be seen as an alternative geometric description of a Fukaya cate-
gory of (M, ω) with L as its only object, a hamiltonian relative version appear-
ing in [Sei]. Using spaces of quilted clusters, we verify, using more general
quilted cluster spaces, that this defines a functor from a homotopy category
of Pin± monotone lagrangian submanifolds hL mono,± (M, ω) to the homotopy
category of cochain complexes hK(Λ-mod) where Λ is an appropriate Novikov
ring.
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Cobordismes lagrangiens et uniréglageLétourneau, Vincent 11 1900 (has links)
Ce mémoire traite de la question suivante: est-ce que les cobordismes lagrangiens préservent l'uniréglage? Dans les deux premiers chapitres, on présente en survol la théorie des courbes pseudo-holomorphes nécessaire. On examine d'abord en détail la preuve que les espaces de courbes $ J $-holomorphes simples est une variété de dimension finie. On présente ensuite les résultats nécessaires à la compactification de ces espaces pour arriver à la définition des invariants de Gromov-Witten. Le troisième chapitre traite ensuite de quelques résultats sur la propriété d'uniréglage, ce qu'elle entraine et comment elle peut être démontrée. Le quatrième chapitre est consacré à la définition et la description de l'homologie quantique, en particulier celle des cobordismes lagrangiens, ainsi que sa structure d'anneau et de module qui sont finalement utilisées dans le dernier chapitre pour présenter quelques cas ou la conjecture tient. / In this dissertation we study the following question: do Lagrangian cobordisms preserve uniruling? In the two first chapters, the necessary pseudoholomorphic curves theory is quickly presented. We first study in detail the proof that the spaces of simple $ J $-holomorphic curves is a manifold of finite dimension. We then present the necessary results to produce the appropriate compactification of these spaces to get to the definition of Gromov-Witten invariants. In the third chapter then some results on the property of uniruling are presented: what are its consequences, how can it be obtained. In the fourth chapter quantum homology is defined, in particular for Lagrangian cobordism, and its ring and module structures are studied which are finally used in the last chapter to present examples of cobordisms which preserves uniruling.
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Fukaya categories of Lagrangian cobordisms and dualityCampling, Emily 11 1900 (has links)
No description available.
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Persistence in discrete Morse theory / Persistenz in der diskreten Morse-TheorieBauer, Ulrich 12 May 2011 (has links)
No description available.
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Source spaces and perturbations for cluster complexesCharest, François 11 1900 (has links)
Dans ce travail, nous définissons des objets composés de disques complexes
marqués reliés entre eux par des segments de droite munis d’une longueur.
Nous construisons deux séries d’espaces de module de ces objets appelés clus-
ters, une qui sera dite non symétrique, la version ⊗, et l’autre qui est dite
symétrique, la version •. Cette construction permet des choix de perturba-
tions pour deux versions correspondantes des trajectoires de Floer introduites
par Cornea et Lalonde ([CL]). Ces choix devraient fournir une nouvelle option
pour la description géométrique des structures A∞ et L∞ obstruées étudiées
par Fukaya, Oh, Ohta et Ono ([FOOO2],[FOOO]) et Cho ([Cho]).
Dans le cas où L ⊂ (M, ω) est une sous-variété lagrangienne Pin± mono-
tone avec nombre de Maslov ≥ 2, nous définissons une structure d’algèbre A∞
sur les points critiques d’une fonction de Morse générique sur L. Cette struc-
ture est présentée comme une extension du complexe des perles de Oh ([Oh])
muni de son produit quantique, plus récemment étudié par Biran et Cornea
([BC]). Plus généralement, nous décrivons une version géométrique d’une
catégorie de Fukaya avec seul objet L qui se veut alternative à la description
(relative) hamiltonienne de Seidel ([Sei]). Nous vérifions la fonctorialité de
notre construction en définissant des espaces de module de clusters occultés
qui servent d’espaces sources pour des morphismes de comparaison. / We define objects made of marked complex disks connected by metric line seg-
ments and construct two sequences of moduli spaces of these objects, referred
as the ⊗ version (nonsymmetric) and the • version (symmetric). This allows
choices of coherent perturbations over the corresponding versions of the Floer
trajectories proposed by Cornea and Lalonde ([CL]). These perturbations are
intended to lead to an alternative geometric description of the (obstructed) A∞
and L∞ structures studied by Fukaya, Oh, Ohta and Ono ([FOOO2],[FOOO])
and Cho ([Cho]).
Given a Pin± monotone lagrangian submanifold L ⊂ (M, ω) with mini-
mal Maslov number ≥ 2, we define an A∞ -algebra structure from the critical
points of a generic Morse function on L. We express this structure as a cochain
complex extending the pearl complex introduced by Oh ([Oh]) and further ex-
plicited by Biran and Cornea ([BC]), equipped with its quantum product. This
could also be seen as an alternative geometric description of a Fukaya cate-
gory of (M, ω) with L as its only object, a hamiltonian relative version appear-
ing in [Sei]. Using spaces of quilted clusters, we verify, using more general
quilted cluster spaces, that this defines a functor from a homotopy category
of Pin± monotone lagrangian submanifolds hL mono,± (M, ω) to the homotopy
category of cochain complexes hK(Λ-mod) where Λ is an appropriate Novikov
ring.
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