DIFFERENTIAL GEOMETRY

Author: FERREIRO PEREZ ROBERTO.
Year: 2003.
University: COMPLUTENSE DE MADRID [www.ucm.es].
Place of defense: FACULTAD DE CIENCIAS MATEMATICAS.
Place of preparation: FACULTAD DE CC. MATEMATICAS.

Summary: In this paper we interpret forms deferenciales grade higher than dimensiÃÂ ³ n of the variety based on fibrado jet of a fibrado as differential forms in a variety of sections overall fibrado, and extends this interpretaciÃÂ ³ n the context of cohomologia equivariante. Applying this construction to the forms features equivariantes in fibrado connections are derived classes cohomologia in space ratio conesiones module trasnformaciones "gauge" asÃÂ structures simplecticas and applications canonical moment. It defines forms pontryagain in fibrado of a variety of metric and applying the canonical structures in the space of matricas module difeomorfismos. In the particular case of dimensiÃÂ ³ n 2, we obtain a form pre-simplectica and aplicaciÃÂ ³ n time, and semuestra that reducciÃÂ ³ n simplectica corresponding space teichmuller with the structure simplectica of weil-petersson.

Author: BRUNA FLORIS LLUÍS.
Year: 2004.
University: AUTÓNOMA DE BARCELONA [www.uab.es].
Place of defense: FACULTAT DE CIÉNCIES.
Place of preparation: ESCUELA DE POSTGRADO.

Summary: The objective of this thesis is to establish the appropriate mathematical framework for the stability of the Einstein equation E = XT, and once this is achieved will find the conditions to give this kind of stability where it is considered a model of Robertson-Walker for the universe. The concept of stability lienal wonder if it really comes to the solutions of an equation linearized used to approximate the solutions of the corresponding linear equation. In the case of the Einstein equation in a vacuum G (g) = 0 this issue gives rise to the classic definition of stability, based on the notion tangencia. In line with the point of view of Einstein in his paper On the gravity waves (1918), the linear stability of G = XT should ensure that a procedure like the following is correct: interpret the universe as a model RW (g0, T0 ), the two connected spor G (g0) = XT0; then given a pertubación T T0 and with the aim of finding an appropriate disturbance * g g0 compliant G (g0 + g *) = X (T0 + T *) , consieramos Dg0 (g *) = X * T. Therefore we need a definition of linear stability adapted to this new situation. Esqeumáticamente, all that is required is that the solutions of the nonlinear equation f (x) = y0 + q those of the equation linearized Dx0f (h) = q can be parameterized by the same vector space. Controlled by the condition **********. Although tangencia has moved to the background, this loose definition has to involve the former when q = 0 (for more dellates see chap. 3). The first is a small modificaicón the second-in fact, the sufficient condition is the same for both: applying differential duty to be thorough and its nucleus should have a supplementary topological. However, this new definition could not be applied directly to the Einstein equation written in the form G = XT because the condition on energy divg (T) = 0 league g and the first strains of T. It must therefore seek a new framework where the variables that represent the geometry and energy or matter are independent. This is achieved through a Cauchy problem as well because then the whole pertubación of data Cauchy it is a fault of the solution and hence the stability linealizaicón of equations ligadrua is equivalent to the system formed by the equations G = XT and divg (T) = 0. Each chapter 1 is entirely devoted to Cauchy problem for the Einstein equation in the presence of matter.

Author: SANTAMARIA MERINO AITOR.
Year: 2004.
University: CARLOS III DE MADRID [www.uc3m.es].
Place of defense: ESCUELA POLITECNICA SUPERIOR.
Place of preparation: UNIVERSIDAD CARLOS III DE MADRID.

Summary: In this thesis is deepened by the description of the Classic Field Theories in terms of Geometry Multisimpléctica. In addition, an analysis of the geometric properties of certain problems in mechanics, interesting for the construction of a new family of digital integrators whose properties convergence exceed those of conventional algorithms. It is hoped in the future to extend field theories. It begins with a brief description of the geometry simpléctica and applications mechanics, accompanied by a parallel exhibition of geometry multisimpléctica and bundles of jets, which are based describing the classical field theories in terms multisimplécticos. In particular, the thesis complete description with new theorems concerning the existence of Darboux coordinates, and an extension of the triple Tulczyjew the Field Theories. It then discusses the concepts of symmetry and conserved quantity equations for the dynamics of the classical theories of fields. Likewise, he examines the dynamics in the presence of Cauchy surfaces. Finally, using properties of the functions generating new numerical methods to obtain geometric, particularizando for mechanics not holónoma and optimal control, with better long-term performance, as evidence on some examples. The thesis concludes with a description of open issues in this area.

Author: PEÑA CARRERA MARTA.
Year: 2004.
University: POLITÉCNICA DE CATALUÑA [www.upc.edu].
Place of defense: AULA CAPELLA ETSEIB.
Place of preparation: U FACULTAT DE MATEMATIQUES I ESTADISTICA SUD.

Author: ALEGRE RUEDA PABLO SEBASTIÁN.
Year: 2004.
University: SEVILLA [www.us.es].
Place of defense: FACULTAD DE MATEMATICAS.
Place of preparation: FACULTAD DE MATEMÁTICAS.

Summary: The Riemann curvature is an important tool in the study of varieties. Thus, it is well known classification spaces constant curvature depending on the value of the curvature. In Geometry casi-Hermítica, F. Tricerri and L. Vanhecke expanded this study spaces sectional holomorfa constant curvature widespread. In this thesis, we introduce the similar case in Geometry casi-contacto metrics, defining spaces curvature phi-seccional continuing widespread. Introducing interesting examples using different geometric tools, such as product warped or alabeados or transformations under and D-conforme of metrics. In addition, we study the fundamental properties of the new spaces defined, paying particular attention to cases that present structures contact metric Sasakianas or trans-Sasakianas. In a second part, we conducted a study of existing inequalities BY. Chen to subarieades an area of curvature phi-seccional continuing widespread, both in the case where such subariedades are tangents to the field structure of the space environment, as if they are normal.

Author: HANS UBER MARÍA BELÉN.
Year: 2004.
University: SEVILLA [www.us.es].
Place of defense: FACULTAD DE MATEMÁTICAS.
Place of preparation: FACULTAD DE MATEMÁTICAS UNIVERSIDAD DE SEVILLA.

Summary: The subvariedades slant are an important type of subvariedades, both in geometry and in the Complex Geometry Contact. These subvariedades are owned constitute a generalization of subvariedades invariants and anti-invariantes, while describing the situation intermediate between the two. Moreover, f-variedades constitute a kind of varieties that embraces the complex and varieties of contact. In this report, presents the definition of subvariedades slant in f-variedades, as a natural extension of the definitions for the complex case and contact, obtaining the first properties, results and characterizations. The study focuses especially in the case where the range environment is a S-variedad or f-variedad with certain very specific conditions. It addresses various aspects which may characterize these subvariedades: the size, the nature of minimalidad and umbilicalidad, curvature, and so on. A case that needs special attention, is that where the size of the smallest possible subvariety is not trivial, where properties are obtained characteristics. Featured are certain types of subvariedades slant in S-variedades, whose definitions are intimately connected with the f-estructura: subvariedades totalmente-f-geodésicas, f-umbilicales and seudo-f-umbilicales. Finally, we get interesting invariant relationship between extrinsic and intrinsic subvariety a slant in a S-variedad with curvature f-seccional constant. Continuously, referred to the examples presented in this report, which aligned the interest of the subject matter.

Author: MARTIN GRILLO RUBEN.
Year: 2006.
University: POLITÉCNICA DE CATALUÑA [www.upc.edu].
Place of defense: SALA D'ACTES DE L'FME.
Place of preparation: U FACULTAT DE MATEMATIQUES I ESTADISTICA SUD.

Summary: This dissertation is devoted to the study of geometric formulations of implicit dynamical systems, and more particularly those that are affine in the highest-order derivative. In the first-order case, these implicit dynamical systems are described by an equation of the form ( ,) Atx Ãx& =b(t,x), where A is a generically singular matrix and b a vector. The main motivation to study this class of systems is that several formalisms of mechanics lead to equations of this kind, amongst them, the most important two: the Lagrangian and Hamiltonian formalisms. A geometric structure to model these implicit dynamical systems is called linearly singular system and has been previously studied for the time-independent case. An implicit dynamical system on a manifold M is modelled by a vector bundle morphism between the tangent bundle TM and another vector bundle E , together with a section of E . It has been shown how various formalisms of time-independent mechanics are included in this framework. As well, procedures (the so-called constraint algorithms) to find the solutions of a linearly singular system has been developed. In this dissertation, these concepts and results of time-independent linearly singular systems are reviewed and the geometric framework is extended in two directions. First, nonholonomic mechanical systems are formulated and studied in terms of linearly singular systems. To perform this, the concepts of subsystem and quotient of a linearly singular system are introduced. Aspects on regularity, consistency and equations of motion are considered. Symmetries and constants of motion of noholonomic systems are also studied. Implicit Hamiltonian systems are also included in this framework. Second, a time-dependent version of the linearly singular systems is given and studied. The main difference with respect to the time-independent case is that affine bundles replace vector bundles, and, instead of the tangent bundle , an appropriate jet bundle is used. Special attention is payed to systems arising from formulations of time-dependent mechanics. A geometric constraint algorithm for time-dependent linearly singular systems is also presented. It is known that a time-dependent differential equation can be considered as an autonomous one by regarding the time as another dependent variable. In this work, a geometrical equivalent of the process of converting a time-dependent differential equation into an autonomous one, that fits well with the given geometrical formulations, is proposed. It is based on the concept of vector hull of an affine space: a vector space that contains the affine space as a proper hyperplane. A review of this topic is given, and some new results about it are presented. The notions and results are extended to deal with affine bundles and vector bundles. Finally, vector hulls of some affine bundles are computed, particularly those of the jet bundles JkM®Jk-1M.