Forme volume

En géométrie différentielle, une forme volume généralise la notion de déterminant aux variétés différentielles. Elle définit une mesure sur la variété, permet le calcul des volumes généralisés, et la définition générale des orientations. Une forme volume se définit comme une forme différentielle de degré maximal, nulle en aucun point. Pour qu'une variété admette une forme volume, il faut et il suffit qu'elle soit orientable. Dans ce cas, il en existe une infinité. En présence d'une structure supplémentaire (riemannienne, symplectique ou autre), il est judicieux de choisir une forme volume spécifique. Généralités Soit M une variété différentielle de dimension n. Une forme volume est une section de la n-ème puissance extérieure \Lambda^n M du fibré cotangent, nulle en aucun point. La régularité de cette section peut dépendre du contexte ; souvent on l'impose de classe C^{\infty}, les résultats pouvant s'adapter

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The group of unimodular automorphisms of a principal bundle and the Euler-Yang-Mills equations

Given a principal bundle G hooked right arrow P -> B (each being compact, connected and oriented) and a G-invariant metric h(P) on P which induces a volume form mu(P), we consider the group of all unimodular automorphisms SAut(P, mu(P)) := {phi is an element of Diff(P) vertical bar phi*mu(P) = mu(P) and phi is G-equivariant) of P, and determines its Euler equation a la Arnold. The resulting equations turn out to be (a particular case of) the Euler-Yang-Mills equations of an incompressible classical charged ideal fluid moving on B. It is also shown that the group SAut(P, mu(P)) is an extension of a certain volume preserving diffeomorphisms group of B by the gauge group Gau(P) of P. (C) 2010 Elsevier B.V. All rights reserved.

2010

Chargement

Some linear and nonlinear problems involving differential forms

Saugata Bandyopadhyay

In this thesis, we study some linear and nonlinear problems involving differential forms. We begin by studying the problem of pullbacks which asks the following question: for two given differential forms, if one is the pullback of the other via a diffeomorphism satisfying some given condition. For volume forms, this problem was studied by Dacorogna-Moser giving a necessary and sufficient condition for the existence of the diffeomorphism with precise regularity. Our goal is to extend this result for general k-forms. We have obtained some necessary and sufficient conditions for two-forms and for some special classes of k-forms with sharp regularity. Then we turn our attention to the problem of differential inclusions involving differential forms. Although for zero-forms, the problem has been extensively studied, essentially nothing was known for higher forms including the curl operator. In this direction, we have obtained some necessary and some sufficient conditions for general k-forms unifying the study of the different cases. Moreover, we show that these necessary and sufficient conditions coincide for k = 1, solving the case of curl operator fairly completely. Besides these problems, we have studied some domain invariance property of the weighted-homogenous and non-homogenous Hardy constants as well. We have showed that the Hardy constant corresponding to these classes of inequalities enjoy, to some extent, the same domain invariance property as that of the Hardy constant corresponding to the standard Hardy's inequality.

EPFL2007

Chargement

L'équation de rappel

Olivier Kneuss

In this thesis we are interested in the following problem : given two differential k–forms g and f, most of the time they will be assumed closed, on what conditions can we pullback g to f by a map φ ? In other words we ask when it is possible to solve the pullback equation φ*(g) = f. This equation originates from differential geometry. This work is divided into five main parts. In Part I we focus on exterior forms. The study of these forms, which is part of the domain of multilinear algebra, will allow us to understand the pullback equation at the algebraic level and will turn out to be a main first step for the study of the problem. Part II deals with the study of differential forms. A special attention will be brought to different versions of Poincaré lemma which will be constantly used all the way through this thesis. In Part III we solve the problem for volume forms k = n. This part, which will use very seldom the machinery of differential geometry, can be read independently of the previous two parts. Part IV sheds light on the case 0 ≤ k ≤ n – 1. Emphasis will be given on the case k = 2. The best results of this present work work will concern 2–forms and volume forms. The cases k = 0, 1, n – 1 are the easiest and will be discussed briefly while for the case 3 ≤ k ≤ n – 2, which will turn much more difficult even already at the algebraic level, we will only obtain results for k-forms with a special structure. Finally, in Part V, we gather numerous useful properties of Hödler spaces. These spaces will be very suitable for the pullback equation ; indeed there are an algebra (contrary to Sobolev spaces in general) and allow to get results with sharp regularity.

EPFL2011

Chargement

Algèbre extérieure

En mathématiques, et plus précisément en algèbre et en analyse vectorielle, l'algèbre extérieure d'un espace vectoriel E est une algèbre associative graduée, notée \Lambda E. La mul

Forme différentielle

En géométrie différentielle, une forme différentielle est la donnée d'un champ d'applications multilinéaires alternées sur les espaces tangents d'une variété différentielle possédant une certaine ré

Géométrie différentielle

vignette|Exemple d'objets étudiés en géométrie différentielle. Un triangle dans une surface de type selle de cheval (un paraboloïde hyperbolique), ainsi que deux droites parallèles. En mathématiques,

PHYS-427: Relativity and cosmology I

Introduce the students to general relativity and its classical tests.

MATH-494: Topics in arithmetic geometry

P-adic numbers are a number theoretic analogue of the real numbers, which interpolate between arithmetics, analysis and geometry. In this course we study their basic properties and give various applications, notably we will prove rationality of the Weil Zeta function.

CIVIL-449: Nonlinear analysis of structures

This course deals with the nonlinear modelling and analysis of structures when subjected to monotonic, cyclic, and dynamic loadings, focusing in particular on the seismic response of structures. It introduces solution methods for nonlinear static and dynamic problems.