Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in De Sitter space

In this paper we give a partially affirmative answer to the following question posed by Haizhong Li: is a complete spacelike hypersurface in De Sitter space S(1)(n+1)(c), n >= 3, with constant normalized scalar curvature R satisfying n-2/nc <= R <= c totally umbilical? (C) 2008 Elsevier B.V. All rights reserved.

Studies on Some Aspects of the Physics of the Early Universe Using Gravitationally Coupled Scalar Field –

This thesis deals with some aspects of the Physics of the early universe, like phase transitions, bubble nucleations and premodial density perturbations which lead to the formation structures in the universe. Quantum aspects of the gravitational interaction play an essential role in retical high-energy physics. The questions of the quantum gravity are naturally connected with early universe and Grand Unification Theories. In spite of numerous efforts, the various problems of quantum gravity remain still unsolved. In this condition, the consideration of different quantum gravity models is an inevitable stage to study the quantum aspects of gravitational interaction. The important role of gravitationally coupled scalar field in the physics of the early universe is discussed in this thesis. The study shows that the scalar-gravitational coupling and the scalar curvature did play a crucial role in determining the nature of phase transitions that took place in the early universe. The key idea in studying the formation structure in the universe is that of gravitational instability.

Curvature Estimates for Submanifolds in Warped Products

We give estimates of the intrinsic and the extrinsic curvature of manifolds that are isometrically immersed as cylindrically bounded submanifolds of warped products. We also address extensions of the results in the case of submanifolds of the total space of a Riemannian submersion.

Sur une classe de structures kählériennes généralisées toriques

Cette thèse concerne le problème de trouver une notion naturelle de «courbure scalaire» en géométrie kählérienne généralisée. L'approche utilisée consiste à calculer l'application moment pour l'action du groupe des difféomorphismes hamiltoniens sur l'espace des structures kählériennes généralisées de type symplectique. En effet, il est bien connu que l'application moment pour la restriction de cette action aux structures kählériennes s'identifie à la courbure scalaire riemannienne. On se limite à une certaine classe de structure kählériennes généralisées sur les variétés toriques notée $DGK_{\omega}^{\mathbb{T}}(M)$ que l'on reconnaît comme étant classifiées par la donnée d'une matrice antisymétrique $C$ et d'une fonction réelle strictement convexe $\tau$ (ayant un comportement adéquat au voisinage de la frontière du polytope moment). Ce point de vue rend évident le fait que toute structure kählérienne torique peut être déformée en un élément non kählérien de $DGK_{\omega}^{\mathbb{T}}(M)$, et on note que cette déformation à lieu le long d'une des classes que R. Goto a démontré comme étant libre d'obstruction. On identifie des conditions suffisantes sur une paire $(\tau,C)$ pour qu'elle donne lieu à un élément de $DGK_{\omega}^{\mathbb{T}}(M)$ et on montre qu'en dimension 4, ces conditions sont également nécessaires. Suivant l'adage «l'application moment est la courbure» mentionné ci-haut, des formules pour des notions de «courbure scalaire hermitienne généralisée» et de «courbure scalaire riemannienne généralisée» (en dimension 4) sont obtenues en termes de la fonction $\tau$. Enfin, une expression de la courbure scalaire riemannienne généralisée en termes de la structure bihermitienne sous-jacente est dégagée en dimension 4. Lorsque comparée avec le résultat des physiciens Coimbra et al., notre formule suggère un choix canonique pour le dilaton de leur théorie.

NEW CHARACTERIZATIONS OF COMPLETE SPACELIKE SUBMANIFOLDS IN SEMI-RIEMANNIAN SPACE FORMS

In this paper we study n-dimensional complete spacelike submanifolds with constant normalized scalar curvature immersed in semi-Riemannian space forms. By extending Cheng-Yau`s technique to these ambients, we obtain results to such submanifolds satisfying certain conditions on both the squared norm of the second fundamental form and the mean curvature. We also characterize compact non-negatively curved submanifolds in De Sitter space of index p.

ENERGY OF GLOBAL FRAMES

The energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T(1) M determined by X. For odd-dimensional spheres, the energy functional has an infimum for each dimension 2k + 1 which is not attained by any non-singular vector field for k > 1. For k = 1, Hopf vector fields are the unique minima. In this paper we show that for any closed Riemannian manifold, the energy of a frame defined on the manifold, possibly except on a finite subset, admits a lower bound in terms of the total scalar curvature of the manifold. In particular, for odd-dimensional spheres this lower bound is attained by a family of frames defined on the sphere minus one point and consisting of vector fields parallel along geodesics.

Black string corrections in variable tension braneworld scenarios

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Conformal invariance of massless Duffin-Kemmer-Petiau theory in Riemannian spacetimes

We investigate the conformal invariance of massless Duffin-Kemmer-Petiau theory coupled to Riemannian spacetimes. We show that, as usual, in the minimal coupling procedure only the spin I sector of the theory - which corresponds to the electromagnetic field - is conformally invariant. We also show that the conformal invariance of the spin 0 sector can be naturally achieved by introducing a compensating term in the Lagrangian. Such a procedure - besides not modifying the spin I sector - leads to the well-known conformal coupling between the scalar curvature and the massless Klein-Gordon-Fock field. Going beyond the Riemannian spacetimes, we briefly discuss the effects of a nonvanishing torsion in the scalar case.

Buracos sônicos em superfícies esféricas

Pós-graduação em Física - IFT

On bifurcation of solutions of the Yamabe problem in product manifolds

We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non-homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds. (C) 2011 Elsevier Masson SAS. All rights reserved.

New characterizations for hyperbolic cylinders in anti-de Sitter spaces

In this paper we study complete maximal spacelike hypersurfaces in anti-de Sitter space H-1(n+1) with either constant scalar curvature or constant non-zero Gauss-Kronecker curvature. We characterize the hyperbolic cylinders H-m(c(1)) x Hn-m(c(2)), 1 <= m <= n - 1, as the only such hypersurfaces with (n - 1) principal curvatures with the same sign everywhere. In particular we prove that a complete maximal spacelike hypersurface in H-1(5) with negative constant Gauss-Kronecker curvature is isometric to H-1(c(1)) x H-3(c(2)). (C) 2012 Elsevier Inc. All rights reserved.

A NOTE ON THE UNIQUENESS OF SOLUTIONS FOR THE YAMABE PROBLEM

Using recent results on the compactness of the space of solutions of the Yamabe problem, we show that in conformal classes of metrics near the class of a nondegenerate solution which is unique (up to scaling) the Yamabe problem has a unique solution as well. This provides examples of a local extension, in the space of conformal classes, of a well-known uniqueness criterion due to Obata.

Equivariant inverse spectral theory and toric orbifolds

Let O-2n be a symplectic toric orbifold with a fixed T-n-action and with a tonic Kahler metric g. In [10] we explored whether, when O is a manifold, the equivariant spectrum of the Laplace Delta(g) operator on C-infinity(O) determines O up to symplectomorphism. In the setting of tonic orbifolds we shmilicantly improve upon our previous results and show that a generic tone orbifold is determined by its equivariant spectrum, up to two possibilities. This involves developing the asymptotic expansion of the heat trace on an orbifold in the presence of an isometry. We also show that the equivariant spectrum determines whether the toric Kahler metric has constant scalar curvature. (C) 2012 Elsevier Inc. All rights reserved.

Heisenberg symmetry and hypermultiplet manifolds

We study the emergence of Heisenberg (Bianchi II) algebra in hyper-Kähler and quaternionic spaces. This is motivated by the rôle these spaces with this symmetry play in N = 2 hypermultiplet scalar manifolds. We show how to construct related pairs of hyper-Kähler and quaternionic spaces under general symmetry assumptions, the former being a zooming-in limit of the latter at vanishing scalar curvature. We further apply this method for the two hyper-Kähler spaces with Heisenberg algebra, which is reduced to U (1) × U (1) at the quaternionic level. We also show that no quaternionic spaces exist with a strict Heisenberg symmetry – as opposed to Heisenberg U (1). We finally discuss the realization of the latter by gauging appropriate Sp(2, 4) generators in N = 2 conformal supergravity.