Abstract. Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Let also A be a smooth symmetrical positive (0, 2)-tensor field in M . By the Sobolev embedding theorem, we can write that there exist K, B > 0 such that for any u ∈ H 2 1 (M ),where H 2 1 (M ) is the standard Sobolev space of functions in L 2 with one derivative in L 2 . We investigate in this paper the value of the sharp K in the equation above, the validity of the corresponding sharp inequality, and the existence of extremal functions for the saturated version of the sharp inequality.Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Also let A be a smooth symmetrical (0, 2)-tensor field in M . In a local chart, A = (A ij ), i, j = 1, . . . , n. We assume that A is positive when acting on 1-forms in the sense that for any x ∈ M , and any η in the cotangent spaceThen, by the Sobolev embedding theorem, we can write that there exist K, B > 0 such that for any u ∈ H 2 1 (M ),where ∇u = (∂ i u) is the 1-form consisting (in local charts) of the first derivatives of u, dv g is the Riemannian volume element of g, and H 2 1 (M ) is the standard Sobolev space consisting of functions in L 2 with one derivative in L 2 . Clearly, the sharp constant B in (0.1) is V −2/n g , where V g is the volume of (M, g), and the corresponding sharp inequality holds true since it holds true for the classical Sobolev inequality [and |∇u| 2 is controled by A x (∇u, ∇u)]. On the other hand, as is easily understood by the fact that A charges some parts of the space M more than others, it is expected that A will affect the sharp constant K in (0.1). Note (0.1) is associated to the operator ∆ g A u = −div g (A x ∇u) which appears in several places in mathematical and physics literature.The questions we ask in this note are: what is the value K s = K s (g) of the sharp K in (0.1), does the corresponding sharp inequality hold true, and if yes, does its saturated version (where B is lowered to its minimum value under the constraint K = K s ) possess extremal functions? When A = g −1 , we are back to the classical problem (dealing with the classical Sobolev inequality). Possible references in book
We propose in this paper a mathematicians' view of the Kaluza-Klein idea of a five dimensional space-time unifying gravitation and electromagnetism, and extension to higher-dimensional space-time. By considering the classification of positive Einstein curvature tensors and the classical Cauchy-Choquet-Bruhat theorems in general relativity, we introduce concepts of types and rigidity. Then, abandoning the usual requirement of a Ricci-flat five dimensional space-time, we show that a unified geometrical frame can be set for gravitation and electromagnetism, giving, by projection on the classical 4-dimensional space-time, the known Einstein-Maxwell-Lorentz equations for charged fluids. Thus, although not introducing, at least at this stage, new physics, we get a very aesthetic presentation of classical physics in the spirit of general relativity. The usual physical concepts, such as mass, energy, charge, trajectory, Maxwell-Lorentz law, are shown to be only various aspects of the geometry, for example curvature, of space-time considered as a Lorentzian manifold; that is no physical objects are introduced in space-time, no laws are given, everything is only geometry. We will then extend this setting to more than 5 dimensions, giving a precise mathematical frame for possible additional physical effects, preserving gravitation and electromagnetism. Version 22 04 2013. 1
We study in this work the existence of minimizing solutions to the critical-power type equation △gu + h.u = f.u n+2 n−2 on a compact riemannian manifold in the limit case normally not solved by variational methods. For this purpose, we use a concept of "critical function" that was originally introduced by E. Hebey and M. Vaugon for the study of second best constant in the Sobolev embeddings. Along the way, we prove an important estimate concerning concentration phenomena's when f is a non-constant function. We give here intuitive details. 1
Around 1920, Kaluza and Klein had the idea to add a fifth dimension to the classical 4-dimensional spacetime of general relativity to create a geometric theory of gravitation and electromagnetism. Today, theoretical evidences, like string theory, suggest the need for a spacetime with more than five dimensions. The mathematical translation of the heuristic idea of a 4-dimensional classical spacetime equipped with extra "small" dimensions, is a fiber bundle structure π : M → M on a (4 + k)dimensional manifold M , with fiber a compact manifold F of dimension k, more shortly a F -fibration.Kaluza and Klein used a fibration with fiber the standard circle S 1 , this fiber carrying the electromagnetic potential. Inclusion of other physical interaction would therefore require a F -fibration with F of the form F = S 1 ×W , W being a compact manifold.However, with such a F -fibration, they is no naturally defined fiber diffeomorphic to S 1 at each point of the manifold, and therefore one looses the possibility to define simply the electromagnetic potential.We want to present in this paper a mathematical structure generalizing the fiber bundle, that enable the possible definition of multiple naturally defined fibers at each point of the manifold, on which therefore one can define objects that depend only on one of the components, S 1 or W , of the global (S 1 × W ) fiber.Although we do not pretend here to model precisely other known physical interactions, we present this geometric structure as a possible way to model or encode deviations from standard 4-dimensional General Relativity, or "dark" effects such as dark matter or energy ; (we refer to the authors' article [3] from which this paper is extracted, but whose purpose is different). Also this geometry was a starting point for the second author's new approach to a geometric unification of general relativity and quantum physics ( see [19]). 1
We propose in this paper a new approach to the Kaluza-Klein idea of a five dimensional spacetime unifying gravitation and electromagnetism, and extension to higher-dimensional space-time. By considering a natural geometric definition of a matter fluid and abandoning the usual requirement of a Ricci-flat five dimensional space-time, we show that a unified geometrical frame can be set for gravitation and electromagnetism, giving, by projection on the classical 4-dimensional space-time, the known Einstein-Maxwell-Lorentz equations for charged fluids. Thus, although not introducing new physics, we get a very aesthetic presentation of classical physics in the spirit of general relativity. The usual physical concepts, such as mass, energy, charge, trajectory, Maxwell-Lorentz law, are shown to be only various aspects of the geometry, for example curvature, of space-time considered as a Lorentzian manifold; that is no physical objects are introduced in space-time, no laws are given, everything is only geometry.We then extend these ideas to more than 5 dimensions, by considering spacetime as a generalization of a (S 1 × W )-fiber bundle, that we named multi-fibers bundle, where S 1 is the circle and W a compact manifold. We will use this geometric structure as a possible way to model or encode deviations from standard 4-dimensional General Relativity, or "dark" effects such as dark matter or energy.This paper is a rewriting and improvement of a previous joint work, [20], with B. Vaugon, M. Dellinger, Z. Faget. 1 1. dF F F = 0, i.e F F F is closed, 2. (∇ ∇ ∇ • F F F ) ♯ = J.
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