The Experts below are selected from a list of 27 Experts worldwide ranked by ideXlab platform
Ugo Boscain - One of the best experts on this subject based on the ideXlab platform.
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the laplace beltrami operator in almost riemannian geometry
Annales de l'Institut Fourier, 2013Co-Authors: Ugo Boscain, Camille LaurentAbstract:Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a Local Orthonormal Frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. Generically tangency points are isolated. In this paper we study the Laplace-Beltrami operator on such a structure. In the case of a compact orientable surface without tangency points, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence a quantum particle in such a structure cannot cross the singular set and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set (i.e. where the vector fields become collinear), all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities. This phenomenon appears also in sub-Riemannian structure which are not equiregular i.e. in which the grow vector depends on the point. We show this fact by analyzing the Martinet
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the laplace beltrami operator in almost riemannian geometry
arXiv: Spectral Theory, 2011Co-Authors: Ugo Boscain, Camille LaurentAbstract:Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a Local Orthonormal Frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. Generically tangency points are isolated. In this paper we study the Laplace-Beltrami operator on such a structure. In the case of a compact orientable surface without tangency points, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence a quantum particle in such a structure cannot cross the singular set and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set (i.e. where the vector fields become collinear), all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities. This phenomenon appears also in sub-Riemannian structure which are not equiregular i.e. in which the grow vector depends on the point. We show this fact by analyzing the Martinet case.
Camille Laurent - One of the best experts on this subject based on the ideXlab platform.
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the laplace beltrami operator in almost riemannian geometry
Annales de l'Institut Fourier, 2013Co-Authors: Ugo Boscain, Camille LaurentAbstract:Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a Local Orthonormal Frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. Generically tangency points are isolated. In this paper we study the Laplace-Beltrami operator on such a structure. In the case of a compact orientable surface without tangency points, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence a quantum particle in such a structure cannot cross the singular set and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set (i.e. where the vector fields become collinear), all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities. This phenomenon appears also in sub-Riemannian structure which are not equiregular i.e. in which the grow vector depends on the point. We show this fact by analyzing the Martinet
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the laplace beltrami operator in almost riemannian geometry
arXiv: Spectral Theory, 2011Co-Authors: Ugo Boscain, Camille LaurentAbstract:Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a Local Orthonormal Frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. Generically tangency points are isolated. In this paper we study the Laplace-Beltrami operator on such a structure. In the case of a compact orientable surface without tangency points, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence a quantum particle in such a structure cannot cross the singular set and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set (i.e. where the vector fields become collinear), all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities. This phenomenon appears also in sub-Riemannian structure which are not equiregular i.e. in which the grow vector depends on the point. We show this fact by analyzing the Martinet case.
Roger Boudet - One of the best experts on this subject based on the ideXlab platform.
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the movement in space time of a Local Orthonormal Frame
2011Co-Authors: Roger BoudetAbstract:The Dirac wave function associated with a particle contains a Lorentz rotation which allows one to deduce a Local moving Frame. This Frame is such that some of its sub-Frames play an important role in the geometrical interpretation of the gauge and the definition of a momentum-energy tensor associated with the particle. What follows is a pure geometrical description of a movement of this Frame and its sub-Frames independently, except the appellation of some entities, of physical considerations.
Eduardo Gallego - One of the best experts on this subject based on the ideXlab platform.
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RUBRIQUE des C.R.A.S.P.: G¶eom¶etrie Difi¶erentielle COURBURE ET CHAMPS DE PLANS
2015Co-Authors: Eduardo GallegoAbstract:R¶esum¶e. Soit M une vari¶et¶e riemanniene orient¶ee munie de deux champs de plans F et H orient¶es, orthogonaux et compl¶ementaires l’un de l’autre. Suivant Albert ([1]) on obtient quelques formules int¶egrales donnant des relations entre la g¶eom¶etrie de la vari¶et¶e d’une part, et celle des champs de plans (courbure, int¶egrabilit¶e et deuxiµeme forme fondamentale), d’autre part. On generalise un resultat de [3] a codimension arbitraire et sans hypothµese d’int¶e-grabilit¶e. CURVATURE AND PLANE FIELDS Abstract. Let M be an oriented Riemannian manifold equipped with two oriented, complementary and orthogonal distributions of planes F and H Following Albert ([1]) we obtain some change integral formulas in the sense that on one side of them there is a term depending on the geometry of the manifold (curvature), and on the other side there are terms depending on the geometry of the plane flelds (curvature, integrability and second fundamental form). We generalize a result from [3] to arbitrary codimension and avoiding the inte-grability condition. I PRELIMINARIES AND NOTATION. Let (M; g) be a Riemannian man-ifold of dimension n, F be a distribution of p-planes and H be the orthogonal distribution with rank q = n ¡ p. We will denote by fe1; : : : ; eng a Local Orthonormal Frame fleld adapted to F, that is, with eA 2 ¡(F) for 1 • A • p and efi 2 ¡(H) for p + 1 • fi • n. Let fµ1; : : : ; µng be the associated dual basis. The connection and curvature forms of the Levi-Civita connection associated with g will be denoted by!ij and ›ij respectively. They satisfy the structural equations: (1) dµi = ¡ nX k=1!ik ^ µk ›ij = d! i j + nX k=1!ik ^!kj By r0 and R0 we mean the covariant derivation symbol and the Riemann tensor of g (cf.[5]). Then, for every X; Y; Z 2 ¡(F) we deflne rXY = hr0XY 1 2 EDUARDO GALLEGO & AGUST¶I REVENT¶OS R(X;Y)Z = rXrY Z ¡ rY rXZ ¡ r[X;Y]Z fi(X;Y) = vr0XY where h is the orthogonal projection of TM to F. It is easy to prove the following \Gauss equation " for a plane flel
Ladu Roberto - One of the best experts on this subject based on the ideXlab platform.
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On some positive mass theorems for Chern-Gauss-Bonnet masses.
'Pisa University Press', 2018Co-Authors: Ladu RobertoAbstract:An asymptotically flat (AF) manifold is a Riemannian manifold that models a spacelike hypersurface in a Lorentzian spacetime obeying the Einstein field equations \begin{equation}\label{EinsteinFieldEquation} \mathcal{E}^g_2 = T \end{equation} (where $\mathcal{E}^g_2 = \mathrm{Ric}_g - \frac 1 2 S_g g$ is the Einstein tensor of $g$ and $T$ is the stress-energy tensor) and representing an \emph{isolated} gravitational system. A prototype example is the Schwarzschild space that models a spherically symmetric static mass distribution, like an idealized sun in an otherwise empty universe. For AF manifolds there is a well established notion of total mass of the system, the ADM mass (named after Arnowitt,Deser and Misner \cite{ADM1, ADM2, ADM3}), that is given as a kind of flux integral. More precisely, if $ (g_{ij} )_{i j} $ are the components of the metric in a suitable coordinate system, the ADM mass is given by (up to a multiplicative constant) \begin{equation} m(g) = \lim_{r\to \infty} \int_{S_r} (g_{ji,i} - g_{ii,j}) \partial_j\ \lrcorner \ dV, \end{equation} where $S_r$ is a coordinate sphere of radius $r$ and $dV$ is the Euclidean volume element. In the case of a Schwarzschildean space, $m(g)$ coincides with the mass of the system $m$. From the definition though, it is not clear whether the ADM mass is positive. The positive mass conjecture (PMC) states that an AF manifold of non negative scalar curvature and suitable decaying condition at infinity has non negative ADM mass. The hypothesis of non negative scalar curvature is not restrictive as it is inherited from the dominant energy condition of the spacetime when the spacelike hypersurface is totally geodesic. Two important results in this direction are due to Schoen and Yau and Witten. The former two authors proved \cite{SchoenYau} the PMC in dimension less than seven, and the latter \cite{Witten} for spin manifolds in any dimensions. Recently \cite{SchoenYau2017} Schoen and Yau announced they have found a proof valid for any dimension, currently their paper is under review. The PMC has played an important role also in Riemannian Geometry, since \cite{SchoenYau} allowed Schoen to deal with the cases that were left open in the famous Yamabe problem, a challenging question arising from conformal geometry. \begin{equation*} \quad \end{equation*} As a matter of fact, Einstein gravity is the simplest example of gravitational theories based on tensor identities of type \begin{equation} \mathcal{L} = T, \end{equation} where $\mathcal{L}$ is a symmetric, divergence-free and natural tensor of the second order (see \cite{Garraffo}). These tensors arise from higher order corrections to the Einstein-Hilbert action in any sensible theory of quantum gravity and are regarded by physicists as some of the most natural generalizations of the Einstein-Hilber action to dimension larger than four (cf. \cite{Camanho}). The tensors $\mathcal{L}$, constitute a vector space that has been exhaustively studied by Lovelock in the 70's, providing a set of generators now known as Lovelock tensors $\{\mathcal{E}_{2k}^g\}_{k\in \mathbb{N}}$. In particular, in $k$-pure Gauss-Bonnet gravities the Einstein field equation \eqref{EinsteinFieldEquation} is replaced by \begin{equation} \mathcal{E}^g_{2k} = T \end{equation} where $\mathcal{E}_{2k}^g$ is the $2k$-Lovelock tensor, given in a Local Orthonormal Frame $\{e_i\}_{i=1,\dots, n}$ by \begin{equation} \mathcal{E}_{2k}^g (e_i, e_j) = -\frac{1}{2^{k+1}} \delta_{j,j_1,\dots, j_{2q}}^{i,i_1,\dots, i_{2q}}R^{i_1,i_2}_{j_1,j_2}\cdots R^{i_{2k-1},i_{2q}}_{j_{2q-1},j_{2q}}, \end{equation} here $R$ is the Riemann curvature tensor of $g$ and $\delta^{M_1,\dots, M_p}_{L_1,\dots, L_p} = \det (\delta_{L_i}^{M_j})_{i,j}$ is the generalized Kronecker delta. The Einstein tensor is then obtained as a particular case when $k=1$. The aim of this thesis was to investigate the definitions of mass $m_k$ for pure Lovelock gravities, focusing on their geometrical interpretation, together with some related positive mass theorems. The analogous of the ADM mass for $k$-pure Gauss-Bonnet gravity is the $k$-Chern-Gauss-Bonnet mass (CBG mass) $m_k$, that includes as a special case for $k=1$ the ADM mass. CGB masses have been introduced only recently, firstly in 2014 for the case $k=2$ by Ge,Wang and Wu \cite{WangWuNewMass} and then in 2017 for general $k$ by the latter two authors \cite{WangWuCGB}. As for the case of the ADM mass, the positivity of the CBG masses has been conjectured ($k$-PMC) but, to the authors knowledge, it has been proved only in two cases: for conformally flat manifolds \cite{GeWangWuConformallyFlat} and for Euclidean graphs \cite{WeiXiong, WangWuNewMass,GiraoSousa}. We aim to give an unified treatment of these masses, discussing several equivalent definitions of them, in particular we worked on a variational characterization of $m_k$ that seems to be new. We also provide proofs, of the $k$-PMC, recently appeared in the literature together with Witten's proof for spin manifold \cite{Witten}. In doing so, we make use of the double forms formalism that allows to efficiently compute curvature invariants and to highlight the symmetry of the relevant geometric quantities