A result from Gromov ensures the existence of a contact structure on any connected non-compact od... more A result from Gromov ensures the existence of a contact structure on any connected non-compact odd dimensional Lie group. But in general such structures are not invariant under left translations. The problem of finding which Lie groups admit a left invariant contact structure (contact Lie groups), is then still wide open. We perform a 'contactization' method to construct, in every odd dimension, many contact Lie groups with a discrete centre and discuss some applications and consequences of such a construction. We give classification results in low dimensions. In any dimension ≥ 7, there are infinitely many locally non-isomorphic solvable contact Lie groups. We also classify contact Lie groups having a prescribed Riemannian or semi-Riemannian structure and derive obstructions results 2 .
HAL (Le Centre pour la Communication Scientifique Directe), Mar 5, 2011
We start by a review of the chronology of mathematical results on the Dirichletto-Neumann map whi... more We start by a review of the chronology of mathematical results on the Dirichletto-Neumann map which paved the way towards the physics of transformational acoustics. We then rederive the expression for the (anisotropic) density and bulk modulus appearing in the pressure wave equation written in the transformed coordinates. A spherical acoustic cloak consisting of an alternation of homogeneous isotropic concentric layers is further proposed based on the effective medium theory. This cloak is characterised by a low reflection and good efficiency over a large bandwidth for both near and far fields, which approximates the ideal cloak with a inhomogeneous and anisotropic distribution of material parameters. The latter suffers from singular material parameters on its inner surface. This singularity depends upon the sharpness of corners, if the cloak has an irregular boundary, e.g. a polyhedron cloak becomes more and more singular when the number of vertices increases if it is star shaped. We thus analyse the acoustic response of a non-singular spherical cloak designed by blowing up a small ball instead of a point, as proposed in [Kohn, Shen, Vogelius, Weinstein, Inverse Problems 24, 015016, 2008]. The multilayered approximation of this cloak requires less extreme densities (especially for the lowest bound). Finally, we investigate another type of non-singular cloaks, known as invisibility carpets [Li and Pendry, Phys. Rev. Lett. 101, 203901, 2008], which mimic the reflection by a flat ground.
Recent static experiments on twist effects in chiral three-dimensional mechanical metamaterials h... more Recent static experiments on twist effects in chiral three-dimensional mechanical metamaterials have been discussed in the context of micropolar Eringen continuum mechanics, which is a generalization of linear Cauchy elasticity. For cubic symmetry, Eringen elasticity comprises nine additional parameters with respect to linear Cauchy elasticity, of which three directly influence chiral effects. Here, we discuss the behavior of the static case of an alternative generalization of linear Cauchy elasticity, the Willis equations. We show that in the homogeneous static cubic case, only one additional parameter with respect to linear Cauchy elasticity results, which directly influences chiral effects. We show that the static Willis equations qualitatively describe the experimentally observed chiral twist effects, too. We connect the behavior to a characteristic length scale.
HAL (Le Centre pour la Communication Scientifique Directe), Jul 20, 2009
We derive the expression for the anisotropic heterogeneous matrices of permittivity and permeabil... more We derive the expression for the anisotropic heterogeneous matrices of permittivity and permeability associated with two-dimensional polygonal and star shaped cloaks. We numerically show using finite elements that the forward scattering worsens when we increase the number of sides in the latter cloaks, whereas it improves for the former ones. This antagonistic behavior is discussed using a rigorous asymptotic approach. We use a symmetry group theoretical approach to derive the cloaks design.
HAL (Le Centre pour la Communication Scientifique Directe), Sep 12, 2012
Plan de l’expose: Introduction sur la refraction negative et le cloaking Partie 1: Focalisation d... more Plan de l’expose: Introduction sur la refraction negative et le cloaking Partie 1: Focalisation des vagues et des ondes de Lamb Partie 2: Capes a vagues, et d’ondes de Lamb dans les plaques Partie 3: Vers le controle des ondes sismiques Conclusion et perspectives
We discuss Cartan-Schouten metrics (Riemannian or pseudo-Riemannian metrics that are parallel wit... more We discuss Cartan-Schouten metrics (Riemannian or pseudo-Riemannian metrics that are parallel with respect to the Cartan-Schouten canonical connection) on
perfect Lie groups. Applications are foreseen in Information Geometry. Throughout this work, the tangent bundle TG and the cotangent bundle T∗G of a Lie group G, are always endowed with their Lie group structures induced by the right trivialization. We show that TG and T∗G are isomorphic if G possesses a biinvariant Riemannian or pseudo-Riemannian metric. We also show that, if on a perfect Lie group, there exists a Cartan-Schouten metric, then it must be biinvariant. We compute all such metrics on the cotangent bundles of simple Lie groups. We further show the following. Endowed with their canonical Lie group structures, the set of unit dual quaternions is isomorphic to T∗SU(2), the set of unit dual split quaternions is isomorphic to T∗SL(2, R). The group SE(3) of special rigid displacements of the Euclidean 3-space is isomorphic to T∗SO(3). The group SE(2, 1) of special rigid displacements of the Minkowski 3-space is isomorphic to T∗SO(2,1). Some results on SE(3) by N. Miolane and X. Pennec, and M. Zefran, V. Kumar and C. Croke, are generalized to SE(2, 1) and to T∗G, for any simple Lie group G.
In this work we propose a strategy based on coordinate transformation to cloak Rayleigh waves. Ra... more In this work we propose a strategy based on coordinate transformation to cloak Rayleigh waves. Rayleigh waves are in-plane elastic waves which propagate along the free surface of semi-infinite media. They are governed by Navier equations that retain their form for an in-plane arbitrary coordinate transformation, upon choosing the specific kinematic relation between displacement fields in virtual, i.e. reference, and transformed, i.e. cloaked, domains. However, the elasticity tensor of the transformed domain is no longer fully symmetric, and thus, it is difficult to design with common materials. Motivated by this issue, we propose a symmetrization technique, based on the arithmetic mean, to obtain anisotropic, yet symmetric, elastic tensors for Rayleigh wave near-cloaking. In particular, by means of time-harmonic numerical simulations and dispersion analyses, we compare the efficiency of triangular and semi-circular cloaks designed with the original non-symmetric tensors and the related symmetrized versions. In addition, different coordinate transformations, e.g. linear, quadratic and cubic, are adopted for the semi-circular cloaks. Through the analyses, we show that a symmetrized semi-circular cloak, obtained upon the use of a quadratic transformation, performs better than the other investigated designs. Our study provides a step towards the design of feasible and efficient broadband elastic metamaterial cloaks for surface waves.
International Journal of Engineering Science, 2023
In this work we propose a strategy based on coordinate transformation to cloak Rayleigh waves. Ra... more In this work we propose a strategy based on coordinate transformation to cloak Rayleigh waves. Rayleigh waves are in-plane elastic waves which propagate along the free surface of semi-infinite media. They are governed by Navier equations that retain their form for an in-plane arbitrary coordinate transformation, upon choosing the specific kinematic relation between displacement fields in virtual, i.e. reference, and transformed, i.e. cloaked, domains. However, the elasticity tensor of the transformed domain is no longer fully symmetric, and thus, it is difficult to design with common materials. Motivated by this issue, we propose a symmetrization technique, based on the arithmetic mean, to obtain anisotropic, yet symmetric, elastic tensors for Rayleigh wave near-cloaking. In particular, by means of time-harmonic numerical simulations and dispersion analyses, we compare the efficiency of triangular and semi-circular cloaks designed with the original non-symmetric tensors and the related symmetrized versions. In addition, different coordinate transformations, e.g. linear, quadratic and cubic, are adopted for the semi-circular cloaks. Through the analyses, we show that a symmetrized semi-circular cloak, obtained upon the use of a quadratic transformation, performs better than the other investigated designs. Our study provides a step towards the design of feasible and efficient broadband elastic metamaterial cloaks for surface waves.
We discuss the classification of 2-solvable Frobenius Lie algebras. We prove that every 2-solvabl... more We discuss the classification of 2-solvable Frobenius Lie algebras. We prove that every 2-solvable Frobenius Lie algebra splits as a semidirect sum of an n-dimensional vector space V and an n-dimensional maximal Abelian subalgebra (MASA) of the full space of endomorphisms of V. We supply a complete classification of 2-solvable Frobenius Lie algebras corresponding to nonderogatory endomorphisms, as well as those given by maximal Abelian nilpotent subalgebras (MANS) of class 2, hence of Kravchuk signature (n-1,0,1). In low dimensions, we classify all 2-solvable Frobenius Lie algebras in general up to dimension 8. We correct and complete the classification list of MASAs of sl(4, R) by Winternitz and Zassenhaus. As a biproduct, we give a simple proof that every nonderogatory endormorphism of a real vector space admits a Jordan form and also provide a new characterization of Cartan subalgebras of sl(n, R).
2018 12th International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials), 2018
Building upon analogies with cloaking of elastic waves in plates, a large scale experiment has de... more Building upon analogies with cloaking of elastic waves in plates, a large scale experiment has demonstrated unprecedented control of surface seismic waves in structured soils. Here, we would like to review recent research advances and remaining challenges in the theory and applications of seismic metamaterials for cloaking and earthquake protection. We recall some results on transformation elastodynamics and introduce mathematical theory of near cloaking for elastic equations. The former is a natural framework for scattering problems in unbounded domains, while the latter addresses boundary measurements in bounded domains. These two fields of investigation bring complementary information on cloaking efficiency. Intimate links between cloaking and wave protection will be also discussed.
Steering waves in elastic solids is more demanding than steering waves in electromagnetism or aco... more Steering waves in elastic solids is more demanding than steering waves in electromagnetism or acoustics. As a result, designing material distributions which are the counterpart of optical invisibility cloaks in elasticity poses a major challenge. Waves of all polarizations should be guided around an obstacle to emerge on the downstream side as though no obstacle were there. Recently, we have introduced the directlattice-transformation approach. This simple and explicit construction procedure led to extremely good cloaking results in the static case. Here, we transfer this approach to the dynamic case, i.e., to elastic waves or phonons. We demonstrate broadband reduction of scattering, with best suppressions exceeding a factor of five when using cubic coordinate transformations instead of linear ones. To reliably and quantitatively test these cloaks efficiency, we use an effective-medium approach.
We investigate the properties of principal elements of Frobenius Lie algebras. We prove that any ... more We investigate the properties of principal elements of Frobenius Lie algebras. We prove that any Lie algebra with a left symmetric algebra structure can be embedded as a subalgebra of some sl(m, K), for K = R or C. Hence, the work of Belavin and Drinfeld on solutions of the Classical Yang-Baxter Equation on simple Lie algebras, applied to the particular case of sl(m, K) alone, paves the way to the complete classification of Frobenius and more generally quasi-Frobenius Lie algebras. We prove that, if a Frobenius Lie algebra has the property that every derivation is an inner derivation, then every principal element is semisimple. As an important case, we prove that in the Lie algebra of the group of affine motions of the Euclidean space of finite dimension, every derivation is inner. We also bring examples of Frobenius Lie algebras that are subalgebras of sl(m, K), but yet have nonsemisimple principal elements as well as some with semisimple principal elements having nonrational eigenvalues, where K = R or C.
We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric ... more We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric with specific properties such as being bi-invariant, flat, negatively curved, Einstein, etc. We classify some of such contact Lie groups and derive some obstruction results to the existence of left invariant contact structures on Lie groups
The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in... more The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in particular to their various important applications. The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in 2D) is the set of pairs of points which contribute to the symmetry set, that is, the closure of the set of pairs of distinct points p and q in M, for which there exists a sphere (resp. a circle) tangent to M at p and at q. The aim of this paper is to address problems related to the smoothness and the singularities of the pre-symmetry sets of 3D shapes. We show that the pre-symmetry set of a smooth surface in 3-space has locally the structure of the graph of a function from R^2 to R^2, in many cases of interest.
We prove that the level sets of a real C^s function of two variables near a non-degenerate critic... more We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We also analyse the cases coming from degenerate critical points, corresponding to elliptic cusps of Gauss on a surface, where the differentiability is now reduced to C^[s/4]. However in all our applications to symmetry sets of families of plane curves, we assume the C^infty smoothness.
In this paper, we consider the intensity surface of a 2D image, we study the evolution of the sym... more In this paper, we consider the intensity surface of a 2D image, we study the evolution of the symmetry sets (and medial axes) of 1-parameter families of iso-intensity curves. This extends the investigation done on 1-parameter families of smooth plane curves (Bruce and Giblin, Giblin and Kimia, etc.) to the general case when the family of curves includes a singular member, as will happen if the curves are obtained by taking plane sections of a smooth surface, at the moment when the plane becomes tangent to the surface. Looking at those surface sections as isophote curves, of the pixel values of an image embedded in the real plane, this allows us to propose to combine object representation using a skeleton or symmetry set representation and the appearance modelling by representing image information as a collection of medial representations for the level-sets of an image.
We discuss the behaviour of vertices and inflexions on one-parameter families of plane curves, wh... more We discuss the behaviour of vertices and inflexions on one-parameter families of plane curves, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a smooth surface by parallel planes. This work is preliminary to an investigation of symmetry sets and medial axes for these families of curves, reported elsewhere.
A result from Gromov ensures the existence of a contact structure on any connected non-compact od... more A result from Gromov ensures the existence of a contact structure on any connected non-compact odd dimensional Lie group. But in general such structures are not invariant under left translations. The problem of finding which Lie groups admit a left invariant contact structure (contact Lie groups), is then still wide open. We perform a 'contactization' method to construct, in every odd dimension, many contact Lie groups with a discrete centre and discuss some applications and consequences of such a construction. We give classification results in low dimensions. In any dimension ≥ 7, there are infinitely many locally non-isomorphic solvable contact Lie groups. We also classify contact Lie groups having a prescribed Riemannian or semi-Riemannian structure and derive obstructions results 2 .
HAL (Le Centre pour la Communication Scientifique Directe), Mar 5, 2011
We start by a review of the chronology of mathematical results on the Dirichletto-Neumann map whi... more We start by a review of the chronology of mathematical results on the Dirichletto-Neumann map which paved the way towards the physics of transformational acoustics. We then rederive the expression for the (anisotropic) density and bulk modulus appearing in the pressure wave equation written in the transformed coordinates. A spherical acoustic cloak consisting of an alternation of homogeneous isotropic concentric layers is further proposed based on the effective medium theory. This cloak is characterised by a low reflection and good efficiency over a large bandwidth for both near and far fields, which approximates the ideal cloak with a inhomogeneous and anisotropic distribution of material parameters. The latter suffers from singular material parameters on its inner surface. This singularity depends upon the sharpness of corners, if the cloak has an irregular boundary, e.g. a polyhedron cloak becomes more and more singular when the number of vertices increases if it is star shaped. We thus analyse the acoustic response of a non-singular spherical cloak designed by blowing up a small ball instead of a point, as proposed in [Kohn, Shen, Vogelius, Weinstein, Inverse Problems 24, 015016, 2008]. The multilayered approximation of this cloak requires less extreme densities (especially for the lowest bound). Finally, we investigate another type of non-singular cloaks, known as invisibility carpets [Li and Pendry, Phys. Rev. Lett. 101, 203901, 2008], which mimic the reflection by a flat ground.
Recent static experiments on twist effects in chiral three-dimensional mechanical metamaterials h... more Recent static experiments on twist effects in chiral three-dimensional mechanical metamaterials have been discussed in the context of micropolar Eringen continuum mechanics, which is a generalization of linear Cauchy elasticity. For cubic symmetry, Eringen elasticity comprises nine additional parameters with respect to linear Cauchy elasticity, of which three directly influence chiral effects. Here, we discuss the behavior of the static case of an alternative generalization of linear Cauchy elasticity, the Willis equations. We show that in the homogeneous static cubic case, only one additional parameter with respect to linear Cauchy elasticity results, which directly influences chiral effects. We show that the static Willis equations qualitatively describe the experimentally observed chiral twist effects, too. We connect the behavior to a characteristic length scale.
HAL (Le Centre pour la Communication Scientifique Directe), Jul 20, 2009
We derive the expression for the anisotropic heterogeneous matrices of permittivity and permeabil... more We derive the expression for the anisotropic heterogeneous matrices of permittivity and permeability associated with two-dimensional polygonal and star shaped cloaks. We numerically show using finite elements that the forward scattering worsens when we increase the number of sides in the latter cloaks, whereas it improves for the former ones. This antagonistic behavior is discussed using a rigorous asymptotic approach. We use a symmetry group theoretical approach to derive the cloaks design.
HAL (Le Centre pour la Communication Scientifique Directe), Sep 12, 2012
Plan de l’expose: Introduction sur la refraction negative et le cloaking Partie 1: Focalisation d... more Plan de l’expose: Introduction sur la refraction negative et le cloaking Partie 1: Focalisation des vagues et des ondes de Lamb Partie 2: Capes a vagues, et d’ondes de Lamb dans les plaques Partie 3: Vers le controle des ondes sismiques Conclusion et perspectives
We discuss Cartan-Schouten metrics (Riemannian or pseudo-Riemannian metrics that are parallel wit... more We discuss Cartan-Schouten metrics (Riemannian or pseudo-Riemannian metrics that are parallel with respect to the Cartan-Schouten canonical connection) on
perfect Lie groups. Applications are foreseen in Information Geometry. Throughout this work, the tangent bundle TG and the cotangent bundle T∗G of a Lie group G, are always endowed with their Lie group structures induced by the right trivialization. We show that TG and T∗G are isomorphic if G possesses a biinvariant Riemannian or pseudo-Riemannian metric. We also show that, if on a perfect Lie group, there exists a Cartan-Schouten metric, then it must be biinvariant. We compute all such metrics on the cotangent bundles of simple Lie groups. We further show the following. Endowed with their canonical Lie group structures, the set of unit dual quaternions is isomorphic to T∗SU(2), the set of unit dual split quaternions is isomorphic to T∗SL(2, R). The group SE(3) of special rigid displacements of the Euclidean 3-space is isomorphic to T∗SO(3). The group SE(2, 1) of special rigid displacements of the Minkowski 3-space is isomorphic to T∗SO(2,1). Some results on SE(3) by N. Miolane and X. Pennec, and M. Zefran, V. Kumar and C. Croke, are generalized to SE(2, 1) and to T∗G, for any simple Lie group G.
In this work we propose a strategy based on coordinate transformation to cloak Rayleigh waves. Ra... more In this work we propose a strategy based on coordinate transformation to cloak Rayleigh waves. Rayleigh waves are in-plane elastic waves which propagate along the free surface of semi-infinite media. They are governed by Navier equations that retain their form for an in-plane arbitrary coordinate transformation, upon choosing the specific kinematic relation between displacement fields in virtual, i.e. reference, and transformed, i.e. cloaked, domains. However, the elasticity tensor of the transformed domain is no longer fully symmetric, and thus, it is difficult to design with common materials. Motivated by this issue, we propose a symmetrization technique, based on the arithmetic mean, to obtain anisotropic, yet symmetric, elastic tensors for Rayleigh wave near-cloaking. In particular, by means of time-harmonic numerical simulations and dispersion analyses, we compare the efficiency of triangular and semi-circular cloaks designed with the original non-symmetric tensors and the related symmetrized versions. In addition, different coordinate transformations, e.g. linear, quadratic and cubic, are adopted for the semi-circular cloaks. Through the analyses, we show that a symmetrized semi-circular cloak, obtained upon the use of a quadratic transformation, performs better than the other investigated designs. Our study provides a step towards the design of feasible and efficient broadband elastic metamaterial cloaks for surface waves.
International Journal of Engineering Science, 2023
In this work we propose a strategy based on coordinate transformation to cloak Rayleigh waves. Ra... more In this work we propose a strategy based on coordinate transformation to cloak Rayleigh waves. Rayleigh waves are in-plane elastic waves which propagate along the free surface of semi-infinite media. They are governed by Navier equations that retain their form for an in-plane arbitrary coordinate transformation, upon choosing the specific kinematic relation between displacement fields in virtual, i.e. reference, and transformed, i.e. cloaked, domains. However, the elasticity tensor of the transformed domain is no longer fully symmetric, and thus, it is difficult to design with common materials. Motivated by this issue, we propose a symmetrization technique, based on the arithmetic mean, to obtain anisotropic, yet symmetric, elastic tensors for Rayleigh wave near-cloaking. In particular, by means of time-harmonic numerical simulations and dispersion analyses, we compare the efficiency of triangular and semi-circular cloaks designed with the original non-symmetric tensors and the related symmetrized versions. In addition, different coordinate transformations, e.g. linear, quadratic and cubic, are adopted for the semi-circular cloaks. Through the analyses, we show that a symmetrized semi-circular cloak, obtained upon the use of a quadratic transformation, performs better than the other investigated designs. Our study provides a step towards the design of feasible and efficient broadband elastic metamaterial cloaks for surface waves.
We discuss the classification of 2-solvable Frobenius Lie algebras. We prove that every 2-solvabl... more We discuss the classification of 2-solvable Frobenius Lie algebras. We prove that every 2-solvable Frobenius Lie algebra splits as a semidirect sum of an n-dimensional vector space V and an n-dimensional maximal Abelian subalgebra (MASA) of the full space of endomorphisms of V. We supply a complete classification of 2-solvable Frobenius Lie algebras corresponding to nonderogatory endomorphisms, as well as those given by maximal Abelian nilpotent subalgebras (MANS) of class 2, hence of Kravchuk signature (n-1,0,1). In low dimensions, we classify all 2-solvable Frobenius Lie algebras in general up to dimension 8. We correct and complete the classification list of MASAs of sl(4, R) by Winternitz and Zassenhaus. As a biproduct, we give a simple proof that every nonderogatory endormorphism of a real vector space admits a Jordan form and also provide a new characterization of Cartan subalgebras of sl(n, R).
2018 12th International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials), 2018
Building upon analogies with cloaking of elastic waves in plates, a large scale experiment has de... more Building upon analogies with cloaking of elastic waves in plates, a large scale experiment has demonstrated unprecedented control of surface seismic waves in structured soils. Here, we would like to review recent research advances and remaining challenges in the theory and applications of seismic metamaterials for cloaking and earthquake protection. We recall some results on transformation elastodynamics and introduce mathematical theory of near cloaking for elastic equations. The former is a natural framework for scattering problems in unbounded domains, while the latter addresses boundary measurements in bounded domains. These two fields of investigation bring complementary information on cloaking efficiency. Intimate links between cloaking and wave protection will be also discussed.
Steering waves in elastic solids is more demanding than steering waves in electromagnetism or aco... more Steering waves in elastic solids is more demanding than steering waves in electromagnetism or acoustics. As a result, designing material distributions which are the counterpart of optical invisibility cloaks in elasticity poses a major challenge. Waves of all polarizations should be guided around an obstacle to emerge on the downstream side as though no obstacle were there. Recently, we have introduced the directlattice-transformation approach. This simple and explicit construction procedure led to extremely good cloaking results in the static case. Here, we transfer this approach to the dynamic case, i.e., to elastic waves or phonons. We demonstrate broadband reduction of scattering, with best suppressions exceeding a factor of five when using cubic coordinate transformations instead of linear ones. To reliably and quantitatively test these cloaks efficiency, we use an effective-medium approach.
We investigate the properties of principal elements of Frobenius Lie algebras. We prove that any ... more We investigate the properties of principal elements of Frobenius Lie algebras. We prove that any Lie algebra with a left symmetric algebra structure can be embedded as a subalgebra of some sl(m, K), for K = R or C. Hence, the work of Belavin and Drinfeld on solutions of the Classical Yang-Baxter Equation on simple Lie algebras, applied to the particular case of sl(m, K) alone, paves the way to the complete classification of Frobenius and more generally quasi-Frobenius Lie algebras. We prove that, if a Frobenius Lie algebra has the property that every derivation is an inner derivation, then every principal element is semisimple. As an important case, we prove that in the Lie algebra of the group of affine motions of the Euclidean space of finite dimension, every derivation is inner. We also bring examples of Frobenius Lie algebras that are subalgebras of sl(m, K), but yet have nonsemisimple principal elements as well as some with semisimple principal elements having nonrational eigenvalues, where K = R or C.
We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric ... more We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric with specific properties such as being bi-invariant, flat, negatively curved, Einstein, etc. We classify some of such contact Lie groups and derive some obstruction results to the existence of left invariant contact structures on Lie groups
The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in... more The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in particular to their various important applications. The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in 2D) is the set of pairs of points which contribute to the symmetry set, that is, the closure of the set of pairs of distinct points p and q in M, for which there exists a sphere (resp. a circle) tangent to M at p and at q. The aim of this paper is to address problems related to the smoothness and the singularities of the pre-symmetry sets of 3D shapes. We show that the pre-symmetry set of a smooth surface in 3-space has locally the structure of the graph of a function from R^2 to R^2, in many cases of interest.
We prove that the level sets of a real C^s function of two variables near a non-degenerate critic... more We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We also analyse the cases coming from degenerate critical points, corresponding to elliptic cusps of Gauss on a surface, where the differentiability is now reduced to C^[s/4]. However in all our applications to symmetry sets of families of plane curves, we assume the C^infty smoothness.
In this paper, we consider the intensity surface of a 2D image, we study the evolution of the sym... more In this paper, we consider the intensity surface of a 2D image, we study the evolution of the symmetry sets (and medial axes) of 1-parameter families of iso-intensity curves. This extends the investigation done on 1-parameter families of smooth plane curves (Bruce and Giblin, Giblin and Kimia, etc.) to the general case when the family of curves includes a singular member, as will happen if the curves are obtained by taking plane sections of a smooth surface, at the moment when the plane becomes tangent to the surface. Looking at those surface sections as isophote curves, of the pixel values of an image embedded in the real plane, this allows us to propose to combine object representation using a skeleton or symmetry set representation and the appearance modelling by representing image information as a collection of medial representations for the level-sets of an image.
We discuss the behaviour of vertices and inflexions on one-parameter families of plane curves, wh... more We discuss the behaviour of vertices and inflexions on one-parameter families of plane curves, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a smooth surface by parallel planes. This work is preliminary to an investigation of symmetry sets and medial axes for these families of curves, reported elsewhere.
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Papers by Andre Diatta
perfect Lie groups. Applications are foreseen in Information Geometry. Throughout this work, the tangent bundle TG and the cotangent bundle T∗G of a Lie group G, are always endowed with their Lie group structures induced by the right trivialization. We show that TG and T∗G are isomorphic if G possesses a biinvariant Riemannian or pseudo-Riemannian metric. We also show that, if on a perfect Lie group, there exists a Cartan-Schouten metric, then it must be biinvariant. We compute all such metrics on the cotangent bundles of simple Lie groups. We further show the following. Endowed with their canonical Lie group structures, the set of unit dual quaternions is isomorphic to T∗SU(2), the set of unit dual split quaternions is isomorphic to T∗SL(2, R). The group SE(3) of special rigid displacements of the Euclidean 3-space is isomorphic to T∗SO(3). The group SE(2, 1) of special rigid displacements of the Minkowski 3-space is isomorphic to T∗SO(2,1). Some results on SE(3) by N. Miolane and X. Pennec, and M. Zefran, V. Kumar and C. Croke, are generalized to SE(2, 1) and to T∗G, for any simple Lie group G.
perfect Lie groups. Applications are foreseen in Information Geometry. Throughout this work, the tangent bundle TG and the cotangent bundle T∗G of a Lie group G, are always endowed with their Lie group structures induced by the right trivialization. We show that TG and T∗G are isomorphic if G possesses a biinvariant Riemannian or pseudo-Riemannian metric. We also show that, if on a perfect Lie group, there exists a Cartan-Schouten metric, then it must be biinvariant. We compute all such metrics on the cotangent bundles of simple Lie groups. We further show the following. Endowed with their canonical Lie group structures, the set of unit dual quaternions is isomorphic to T∗SU(2), the set of unit dual split quaternions is isomorphic to T∗SL(2, R). The group SE(3) of special rigid displacements of the Euclidean 3-space is isomorphic to T∗SO(3). The group SE(2, 1) of special rigid displacements of the Minkowski 3-space is isomorphic to T∗SO(2,1). Some results on SE(3) by N. Miolane and X. Pennec, and M. Zefran, V. Kumar and C. Croke, are generalized to SE(2, 1) and to T∗G, for any simple Lie group G.