M. L. Labbi - Academia.edu (original) (raw)

Papers by M. L. Labbi

We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone tha... more We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone that were obtained in the mixed exterior algebra. In particular, we introduce a second product in the space of double forms, namely the composition product, which provides this space with a second associative algebra structure. The composition product interacts with the exterior product of double forms; the resulting relations provide simple alternative proofs to some classical linear algebra identities as well as to recent results in the exterior algebra of double forms. We define a refinement of the notion of pure curvature of Maillot and we use one of the basic identities to prove that if a Riemannian n-manifold has k-pure curvature and n ≥ 4k then its Pontrjagin class of degree 4k vanishes.

Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is... more Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total (2k)-th Gauss-Bonnet curvature function, called (2k)-minimal submanifolds. We prove that they are characterized by the vanishing of a higher mean curvature, namely the (2k + 1)-Gauss-Bonnet curvature. Furthermore, we show that several properties of usual minimal submanifolds can be naturally generalized to (2k)-minimal submanifolds.

Transactions of the American Mathematical Society, 2005

We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra ... more We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the (p, q)-curvatures. They are a generalization of the p-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for p = 0, the (0, q)curvatures coincide with the H. Weyl curvature invariants, for p = 1 the (1, q)curvatures are the curvatures of generalized Einstein tensors, and for q = 1 the (p, 1)-curvatures coincide with the p-curvatures. Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension n ≥ 4, and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.

The usual Einstein metrics are those for which the first Ricci contraction of the covariant Riema... more The usual Einstein metrics are those for which the first Ricci contraction of the covariant Riemann curvature tensor is proportional to the metric. Assuming the same type of restrictions but instead on the different contractions of the generalized covariant Gauss-Kronecker tensors Rp, leads to several generalizations of Einstein’s condition. In this paper, we treat some properties of these metrics. Mathematics Subject Classification (2000). 53C25, 58E11.

In this paper we study positive Einstein curvature which is a condition on the Riemann curvature ... more In this paper we study positive Einstein curvature which is a condition on the Riemann curvature tensor intermediate between positive scalar curvature and positive sec- tional curvature. We prove some constructions and obstructions for positive Einstein curva- ture on compact manifolds generalizing similar well known results for the scalar curvature. Finally, because our problem is relatively new, many open questions are included.

Proceedings of the American Mathematical Society, 1999

In this paper we construct new Riemannian metrics with positive isotropic curvature on compact ma... more In this paper we construct new Riemannian metrics with positive isotropic curvature on compact manifolds which fiber over the circle. We also study the relationship between the positivity of the isotropic curvature and the positivity of the p p -curvature.

We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone tha... more We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone that were obtained in the mixed exterior algebra. In particular, we introduce a second product in the space of double forms, namely the composition product, which provides this space with a second associative algebra structure. The composition product interacts with the exterior product of double forms; the resulting relations provide simple alternative proofs to some classical linear algebra identities as well as to recent results in the exterior algebra of double forms.\\ We define a refinement of the notion of pure curvature of Maillot and we use one of the basic identities to prove that if a Riemannian nnn-manifold has kkk-pure curvature and ngeq4kn\geq 4kngeq4k then its Pontrjagin class of degree 4k4k4k vanishes.

Symmetry, Integrability and Geometry: Methods and Applications, 2007

The (2k)-th Gauss-Bonnet curvature is a generalization to higher dimensions of the (2k)-dimension... more The (2k)-th Gauss-Bonnet curvature is a generalization to higher dimensions of the (2k)-dimensional Gauss-Bonnet integrand, it coincides with the usual scalar curvature for k = 1. The Gauss-Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where they are known as the Lagrangian of Lovelock gravity, Gauss-Bonnet Gravity and Lanczos gravity. In this paper we present various aspects of these curvature invariants and review their variational properties. In particular, we discuss natural generalizations of the Yamabe problem, Einstein metrics and minimal submanifolds.

Differential Geometry and its Applications, 2002

In this paper we give a new proof of Micallef-Wang result concerning the stability of positive is... more In this paper we give a new proof of Micallef-Wang result concerning the stability of positive isotropic curvature under surgeries in codimension n (n is the dimension of the manifold in question). And we show that this property is not true in codimensions n − 1. We also prove that any finitely presented group can be the fundamental group of a compact n-manifold with positive p-curvature (p n − 4).

Calculus of Variations and Partial Differential Equations, 2007

The Gauss-Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss-Bonn... more The Gauss-Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension 2k, as the scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals on the space of all Riemannian metrics on the manifold under consideration. An important property of this derivative is that it depends only on the curvature tensor and not on its covariant derivatives. We show that the critical points of this functional once restricted to metrics with unit volume are generalized Einstein metrics and once restricted to a pointwise conformal class of metrics are metrics with constant Gauss-Bonnet curvature.

The second H. Weyl curvature invariant of a Riemannian manifold, denoted h 4 , is the second curv... more The second H. Weyl curvature invariant of a Riemannian manifold, denoted h 4 , is the second curvature invariant which appears in the well known tube formula of H. Weyl. It coincides with the Gauss-Bonnet integrand in dimension 4. A crucial property of h 4 is that it is nonnegative for Einstein manifolds, hence it provides a geometric obstruction to the existence of Einstein metrics in dimensions ≥ 4, independently from the sign of the Einstein constant. This motivates our study of the positivity of this invariant. Here in this paper we prove many constructions of metrics with positive second H. Weyl curvature invariant, generalizing similar well known results for the scalar curvature.

Cet article est le texte de mon habilitationà diriger des recherches. Onétudie différentes notion... more Cet article est le texte de mon habilitationà diriger des recherches. Onétudie différentes notions de courbure riemanniennes : la p-courbure, qui interpole entre courbure scalaire et courbure sectionnelle, les courbures de Gauss-Bonnet-Weyl qui constituent une autre interpolation allant de la courbure scalaire jusqu'à l'intégrand de Gauss-Bonnet. Les (p, q)-courbures que nous dégageons englobent toutes ces notions. On examine ensuite le terme en courbure de la formule classique de Weitzenböck. Onétudie aussi les propriétés de positivité de la p-courbure, la seconde courbure de Gauss-Bonnet-Weyl, la courbure d'Einstein et de la courbure isotrope.

We use the ring of curvature structures, to answer a question of M. Berger about a variational fo... more We use the ring of curvature structures, to answer a question of M. Berger about a variational formula for the H. Weyl curvature invariants. Recall that the (2k)-H. Weyl curvature invariant is a polynomial of degree 2k with respect to the Riemannian curvature. In dimension 2k it is just the Gauss-Bonnet integrand, in higher dimensions it appears naturally, as an integrand, in the well known H. Weyl's tube formula.

During an operation of surgery on a Riemannian manifold and along a given embedded submanifold, (... more During an operation of surgery on a Riemannian manifold and along a given embedded submanifold, (see [1, 2, 3]), one needs to replace the (old) metric induced by the exponential map on a tubular neighborhood of the submanifold by the Sasakian metric. So a good understanding of the behavior of these two metrics is important, this is our main goal in this paper. In particular, we prove that these two metrics are tangent up to the order one if and only if the submanifold is totally geodesic. In the case where the ambient space is an Euclidean space, we prove that the difference of these two metrics is quadratic in the radius of the tube and depends only on the second fundamental form of the submanifold. Also the case of spherical and hyperbolic space forms are studied.

Arxiv preprint arXiv:0807.2058, 2008

This is a paper based on a talk given at the conference on Conformal Geometry which held at Rosco... more This is a paper based on a talk given at the conference on Conformal Geometry which held at Roscoff in France in the 2008 summer. We study some aspects of the equation arising from the problem of the existence on a given closed Riemannian manifold of dimension n ≥ 4 , of a conformal metric with constant h 4 curvature. We establish a simple formula relating the second Gauss-Bonnet curvature h 4 to the σ 2 curvature and we study some positivity properties of these two quadratic curvatures. We use different quadratic curvatures to characterize space forms, Einstein metrics and conformally flat metrics. In the appendix we introduce natural generalizations of Newton transformations, the corresponding Newton identities are used to obtain Avez type formulas for all the Gauss-Bonnet curvatures. Contents 1. Introduction 2. Gauss-Bonnet Curvatures, σ k Curvatures and the h 2k Yamabe Problem 2.1. Preliminaries 2.2. Gauss-Bonnet curvatures 2.3. Einstein-Lovelock tensors 2.4. The σ k curvatures 2.5. The h 2k Yamabe problem 3. Quadratic scalar curvatures 3.1. Positivity properties of h 4 and σ 2 4. The h 4-Yamabe Equation 4.1. Differential operators of Laplace type

Geometriae Dedicata, 2004

In this paper we study positive Einstein curvature which is a condition on the Riemann curvature ... more In this paper we study positive Einstein curvature which is a condition on the Riemann curvature tensor intermediate between positive scalar curvature and positive sectional curvature. We prove some constructions and obstructions for positive Einstein curvature on compact manifolds generalizing similar well known results for the scalar curvature. Finally, because our problem is relatively new, many open questions are included.

We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone tha... more We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone that were obtained in the mixed exterior algebra. In particular, we introduce a second product in the space of double forms, namely the composition product, which provides this space with a second associative algebra structure. The composition product interacts with the exterior product of double forms; the resulting relations provide simple alternative proofs to some classical linear algebra identities as well as to recent results in the exterior algebra of double forms. We define a refinement of the notion of pure curvature of Maillot and we use one of the basic identities to prove that if a Riemannian n-manifold has k-pure curvature and n ≥ 4k then its Pontrjagin class of degree 4k vanishes.

Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is... more Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total (2k)-th Gauss-Bonnet curvature function, called (2k)-minimal submanifolds. We prove that they are characterized by the vanishing of a higher mean curvature, namely the (2k + 1)-Gauss-Bonnet curvature. Furthermore, we show that several properties of usual minimal submanifolds can be naturally generalized to (2k)-minimal submanifolds.

Transactions of the American Mathematical Society, 2005

We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra ... more We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the (p, q)-curvatures. They are a generalization of the p-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for p = 0, the (0, q)curvatures coincide with the H. Weyl curvature invariants, for p = 1 the (1, q)curvatures are the curvatures of generalized Einstein tensors, and for q = 1 the (p, 1)-curvatures coincide with the p-curvatures. Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension n ≥ 4, and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.

The usual Einstein metrics are those for which the first Ricci contraction of the covariant Riema... more The usual Einstein metrics are those for which the first Ricci contraction of the covariant Riemann curvature tensor is proportional to the metric. Assuming the same type of restrictions but instead on the different contractions of the generalized covariant Gauss-Kronecker tensors Rp, leads to several generalizations of Einstein’s condition. In this paper, we treat some properties of these metrics. Mathematics Subject Classification (2000). 53C25, 58E11.

In this paper we study positive Einstein curvature which is a condition on the Riemann curvature ... more In this paper we study positive Einstein curvature which is a condition on the Riemann curvature tensor intermediate between positive scalar curvature and positive sec- tional curvature. We prove some constructions and obstructions for positive Einstein curva- ture on compact manifolds generalizing similar well known results for the scalar curvature. Finally, because our problem is relatively new, many open questions are included.

Proceedings of the American Mathematical Society, 1999

In this paper we construct new Riemannian metrics with positive isotropic curvature on compact ma... more In this paper we construct new Riemannian metrics with positive isotropic curvature on compact manifolds which fiber over the circle. We also study the relationship between the positivity of the isotropic curvature and the positivity of the p p -curvature.

We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone tha... more We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone that were obtained in the mixed exterior algebra. In particular, we introduce a second product in the space of double forms, namely the composition product, which provides this space with a second associative algebra structure. The composition product interacts with the exterior product of double forms; the resulting relations provide simple alternative proofs to some classical linear algebra identities as well as to recent results in the exterior algebra of double forms.\\ We define a refinement of the notion of pure curvature of Maillot and we use one of the basic identities to prove that if a Riemannian nnn-manifold has kkk-pure curvature and ngeq4kn\geq 4kngeq4k then its Pontrjagin class of degree 4k4k4k vanishes.

Symmetry, Integrability and Geometry: Methods and Applications, 2007

The (2k)-th Gauss-Bonnet curvature is a generalization to higher dimensions of the (2k)-dimension... more The (2k)-th Gauss-Bonnet curvature is a generalization to higher dimensions of the (2k)-dimensional Gauss-Bonnet integrand, it coincides with the usual scalar curvature for k = 1. The Gauss-Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where they are known as the Lagrangian of Lovelock gravity, Gauss-Bonnet Gravity and Lanczos gravity. In this paper we present various aspects of these curvature invariants and review their variational properties. In particular, we discuss natural generalizations of the Yamabe problem, Einstein metrics and minimal submanifolds.

Differential Geometry and its Applications, 2002

In this paper we give a new proof of Micallef-Wang result concerning the stability of positive is... more In this paper we give a new proof of Micallef-Wang result concerning the stability of positive isotropic curvature under surgeries in codimension n (n is the dimension of the manifold in question). And we show that this property is not true in codimensions n − 1. We also prove that any finitely presented group can be the fundamental group of a compact n-manifold with positive p-curvature (p n − 4).

Calculus of Variations and Partial Differential Equations, 2007

The Gauss-Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss-Bonn... more The Gauss-Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension 2k, as the scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals on the space of all Riemannian metrics on the manifold under consideration. An important property of this derivative is that it depends only on the curvature tensor and not on its covariant derivatives. We show that the critical points of this functional once restricted to metrics with unit volume are generalized Einstein metrics and once restricted to a pointwise conformal class of metrics are metrics with constant Gauss-Bonnet curvature.

The second H. Weyl curvature invariant of a Riemannian manifold, denoted h 4 , is the second curv... more The second H. Weyl curvature invariant of a Riemannian manifold, denoted h 4 , is the second curvature invariant which appears in the well known tube formula of H. Weyl. It coincides with the Gauss-Bonnet integrand in dimension 4. A crucial property of h 4 is that it is nonnegative for Einstein manifolds, hence it provides a geometric obstruction to the existence of Einstein metrics in dimensions ≥ 4, independently from the sign of the Einstein constant. This motivates our study of the positivity of this invariant. Here in this paper we prove many constructions of metrics with positive second H. Weyl curvature invariant, generalizing similar well known results for the scalar curvature.

Cet article est le texte de mon habilitationà diriger des recherches. Onétudie différentes notion... more Cet article est le texte de mon habilitationà diriger des recherches. Onétudie différentes notions de courbure riemanniennes : la p-courbure, qui interpole entre courbure scalaire et courbure sectionnelle, les courbures de Gauss-Bonnet-Weyl qui constituent une autre interpolation allant de la courbure scalaire jusqu'à l'intégrand de Gauss-Bonnet. Les (p, q)-courbures que nous dégageons englobent toutes ces notions. On examine ensuite le terme en courbure de la formule classique de Weitzenböck. Onétudie aussi les propriétés de positivité de la p-courbure, la seconde courbure de Gauss-Bonnet-Weyl, la courbure d'Einstein et de la courbure isotrope.

We use the ring of curvature structures, to answer a question of M. Berger about a variational fo... more We use the ring of curvature structures, to answer a question of M. Berger about a variational formula for the H. Weyl curvature invariants. Recall that the (2k)-H. Weyl curvature invariant is a polynomial of degree 2k with respect to the Riemannian curvature. In dimension 2k it is just the Gauss-Bonnet integrand, in higher dimensions it appears naturally, as an integrand, in the well known H. Weyl's tube formula.

During an operation of surgery on a Riemannian manifold and along a given embedded submanifold, (... more During an operation of surgery on a Riemannian manifold and along a given embedded submanifold, (see [1, 2, 3]), one needs to replace the (old) metric induced by the exponential map on a tubular neighborhood of the submanifold by the Sasakian metric. So a good understanding of the behavior of these two metrics is important, this is our main goal in this paper. In particular, we prove that these two metrics are tangent up to the order one if and only if the submanifold is totally geodesic. In the case where the ambient space is an Euclidean space, we prove that the difference of these two metrics is quadratic in the radius of the tube and depends only on the second fundamental form of the submanifold. Also the case of spherical and hyperbolic space forms are studied.

Arxiv preprint arXiv:0807.2058, 2008

This is a paper based on a talk given at the conference on Conformal Geometry which held at Rosco... more This is a paper based on a talk given at the conference on Conformal Geometry which held at Roscoff in France in the 2008 summer. We study some aspects of the equation arising from the problem of the existence on a given closed Riemannian manifold of dimension n ≥ 4 , of a conformal metric with constant h 4 curvature. We establish a simple formula relating the second Gauss-Bonnet curvature h 4 to the σ 2 curvature and we study some positivity properties of these two quadratic curvatures. We use different quadratic curvatures to characterize space forms, Einstein metrics and conformally flat metrics. In the appendix we introduce natural generalizations of Newton transformations, the corresponding Newton identities are used to obtain Avez type formulas for all the Gauss-Bonnet curvatures. Contents 1. Introduction 2. Gauss-Bonnet Curvatures, σ k Curvatures and the h 2k Yamabe Problem 2.1. Preliminaries 2.2. Gauss-Bonnet curvatures 2.3. Einstein-Lovelock tensors 2.4. The σ k curvatures 2.5. The h 2k Yamabe problem 3. Quadratic scalar curvatures 3.1. Positivity properties of h 4 and σ 2 4. The h 4-Yamabe Equation 4.1. Differential operators of Laplace type

Geometriae Dedicata, 2004

In this paper we study positive Einstein curvature which is a condition on the Riemann curvature ... more In this paper we study positive Einstein curvature which is a condition on the Riemann curvature tensor intermediate between positive scalar curvature and positive sectional curvature. We prove some constructions and obstructions for positive Einstein curvature on compact manifolds generalizing similar well known results for the scalar curvature. Finally, because our problem is relatively new, many open questions are included.