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Mohammed Larbi Labbi

Publications and source records attributed to Mohammed Larbi Labbi.

14 recordsLinked to original sources
arXiv2026

On some critical Riemannian metrics and Thorpe-type conditions

We study critical metrics of higher-order curvature functionals on compact Riemannian $n$-manifolds $(M,g)$. For an integer $k$ with $2 \leq 2k \leq n$, let $R^k$ denote the $k$-th exterior power of the Riemann curvature tensor. We investigate the Riemannian functionals \[H_{2k}(g)=\int_M \operatorname{tr}(R^k)\,\mathrm{dvol}_g\quad\text{and}\quad G_{2k}(g)=\int_M \|R^k\|^2\,\mathrm{dvol}_g,\] which generalize the Hilbert--Einstein functional and the total squared norm curvature, obtained for $k=1$ respectively. Using the formalism of double forms, we develop a systematic variational framework yielding compact first variation formulas for these functionals. Two key lemmas streamline the variational computations. A central technical ingredient is a generalization of the classical Lanczos identity to symmetric double forms of arbitrary even degree, providing explicit algebraic relations between the tensors $\cc^{2k-1}(R^k \circ R^k)$ and $\cc^{4k-1}(R^{2k})$. As a main geometric application, we introduce $(2k)$-Thorpe and $(2k)$-anti-Thorpe metrics, defined by self-duality and anti-self-duality conditions on $g^{r-2k}R^k$ in even dimensions $n=2r$. In the critical dimension $n=4k$, these metrics are absolute minimizers of $G_{2k}$, with the minimum determined by the Euler characteristic. For $n>4k$, they satisfy a harmonicity property leading to rigidity results under suitable curvature positivity assumptions. We further establish equivalences among variational criticality conditions. For hyper-$(2k)$-Einstein metrics, characterized by $\cc R^k=λg^{2k-1}$, being critical for $G_{2k}$ is equivalent to being $(4k)$-Einstein and to being weakly $(2k)$-Einstein. In the locally conformally flat setting, we classify all $4$-Thorpe metrics, showing that they are either space forms or Riemannian products $\mathbb{S}^r(c) \times \mathbb{H}^r(-c)$.

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arXiv2026

Generalized Double Duals of the Riemann Tensor in Geometry and Gravity

The Riemann curvature tensor fully encodes local geometry, but its Ricci contraction retains only limited information: only the Ricci tensor and the scalar curvature survive, while the Weyl curvature vanishes identically. We show that contracting instead the double dual of the Riemann tensor unlocks the full curvature structure, producing a canonical hierarchy of symmetric, divergence--free $(p,p)$ double forms. These tensors satisfy the first Bianchi identity and obey a hereditary contraction relation interpolating between the double dual tensor and the Einstein tensor. We prove that, in a generic geometric setting, each tensor in this hierarchy is the unique divergence--free $(p,p)$ double form depending linearly on the Riemann curvature tensor, thereby providing canonical higher--rank parents of the Einstein tensor. Their sectional curvatures coincide with the $p$--curvatures; notably, the $2$--curvature determines the full Riemann curvature tensor and forces the $\hat A$--genus of a compact spin manifold to vanish when nonnegative, a property not shared by Ricci or scalar curvature. Finally, we extend the construction to Gauss--Kronecker curvature tensors and Lovelock theory, showing in particular that the second Lovelock tensor $T_4$ admits a genuine four--index parent tensor.

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arXiv2024

Hodge-de Rham and Lichnérowicz Laplacians on double forms and some vanishing theorems

A $(p,q)$-double form on a Riemannian manifold $(M,g)$ can be considered simultaneously as a vector-valued differential $p$-form over $M$ or alternatively as a vector-valued $q$-form. Accordingly, the usual Hodge-de Rham Laplacian on differential forms can be extended to double forms in two ways. The differential operators obtained in this way are denoted by $Δ$ and $\widetildeΔ$.\\ In this paper, we show that the Lichnérowicz Laplacian $Δ_L$ once operating on double forms, is nothing but the average of the two operators mentioned above. We introduce a new product on double forms to establish index-free formulas for the curvature terms in the Weitzenböck formulas corresponding to the Laplacians $Δ, \widetildeΔ$ and $Δ_L$. We prove vanishing theorems for the Hodge-de Rham Laplacian $Δ$ on $(p,0)$ double forms and for $Δ_L$ and $Δ$ on symmetric double forms of arbitrary order. These results generalize recent results by Petersen-Wink. Our vanishing theorems reveal the impact of the role played by the rank of the eigenvectors of the curvature operator on the structure (e.g. the topology) of the manifold.

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arXiv2023

On a stratification of positive scalar curvature compact manifolds

For a compact PSC Riemannian $n$-manifold $(M,g)$, the metric constant $\mathrm {Riem}(g)\in (0, \binom{n}{2}]$ is defined to be the infinimum over $M$ of the spectral scalar curvature $\frac{\sum_{i=1}^Nλ_i}{λ_{\rm max}}$ of $g$, where $λ_1, ...,λ_N$ are the eigenvalues of the curvature operator of $g$ and $λ_{\rm max}$ is the maximal eigenvalue. The functional $g\to \mathrm {Riem}(g)$ is continuous, re-scale invariant and defines a stratification of the space of PSC metrics over $M$. We introduce as well the smooth constant $\mathbf {Riem}(M)\in (0, \binom{n}{2}]$, which is the supremum of $\mathrm {Riem}(g)$ over the set of all psc Riemannian metrics $g$ on $M$. \\ In this paper, we show that in the top layer, compact manifolds with $\mathbf{Riem}=\binom{n}{2}$ are positive space forms. No manifolds have their $\mathbf{Riem}$ in the interval $(\binom{n}{2}-2, \binom{n}{2})$. The manifold $S^{n-1}\times S^1$ and arbitrary connected sums of copies of it with connected sums of positive space forms all have $\mathbf{Riem}=\binom{n-1}{2}$. For $1\leq p\leq n-2\leq 5$, we prove that the manifolds $S^{n-p}\times T^p$ take the intermediate values $\mathbf {Riem}=\binom{n-p}{2}$. From the bottom, we prove that simply connected (resp. $2$-connected, $3$-connected and non-string) compact manifolds of dimension $\geq 5$ (resp. $\geq 7$, $\geq 9$) have $\mathbf{Riem}\geq 1$ (resp. $\geq 3$, $\geq 6$). The proof of these last three results is based on surgery. In fact, we prove that the smooth $\mathbf{Riem}$ constant doesn't decrease after a surgery on the manifold with adequate codimension.

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arXiv2022

On the conformal Ein invariants

For a compact Riemannian $n$-manifold $(M,g)$ of positive scalar curvature, the capital $\Ein$ invariant of $g$ is defined to be the infinimum over $M$ of the quotient of the scalar curvature by the maximal eigenvalue of the Ricci curvature. This is a re-scale invariant and belongs to the interval $(0,n]$. For a positive conformal class $[g]$, we define the conformal invariant $\Ein([g]):=\sup\{\Ein(g): g\in [g]\}$. In this paper, we prove vanihing theorems for Betti numbers and for the higher homotopy groups of $M$ under optimal lower bounds on $\Ein([g])$ assuming that $g$ is locally conformally flat. We establish an inequality relating our invariant to Schoen-Yau conformal invariant $d(M,[g])$ from which we deduce a classification result for locally conformally flat manifolds with higher $\Ein([g])$. We show that the class of locally conformally flat manifolds with $\Ein([g])>k$ is stable under the operation of connected sums for $0<k<n-1.$\\ For a general positive conformal class, we prove in dimension $4$ an inequality relating $\Ein([g])$ to the first and second Yamabe invariants. Similar results are proved in this paper for an analogous conformal invariant, namely the small $\ein$ invariant.

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arXiv2020

On modified Einstein tensors and two smooth invariants of compact manifolds

Let $(M,g)$ be a Riemannian $n$-manifold, we denote by $\Ric$ and $\Scal$ the Ricci and the scalar curvatures of $g$. For scalars $k<n$, the modified Einstein tensors denoted $\Eink$ are defined as $\Eink :=\Scal \, g -k\Ric$. Note that the usual Einstein tensor coincides with the half of $\Eint$ and ${\rm Ein}_0=\Scal.g$. It turns out that all these new modified tensors, for $0<k<n$, are still gradients of the total scalar curvature functional but with respect to modified integral scalar products. In this paper we study the positivity properties of these tensors that generalize the positivity properties of the scalar curvature ($k=0$) and positive Einstein curvature ($k=2$). The positivity of $\Eink$ for some positive $k$ implies the positivity of all ${\rm Ein}_l$ with $0\leq l\leq k$ and so we define a smooth invariant $\cEin(M)$ of $M$ to be the supremum of positive k's that renders $\Eink$ positive. By definition $\cEin(M)\in [0,n]$, it is zero if and only if $M$ has no positive scalar curvature metrics and it is maximal equal to $n$ if $M$ possesses an Einstein metric with positive scalar curvature. In some sense, $\cEin(M)$ measures how far is $M$ to admit an Einstein metric of positive scalar curvature. In this paper we prove that $\cEin(M)\geq 2$ if $M$ admits an effective action by a non abelian connected Lie group or if $M$ is simply connected of positive scalar curvature and dimension $\geq 5$. We prove as well that the invariant $\cEin$ increases after a surgery operation on the manifold $M$ or by assuming that the manifold $M$ has higher connectivity. We prove that the condition $\cEin(M)\leq n-2$ does not imply any restriction on the first fundamental group of $M$. We define and prove similar properties for an analogous invariant namely $\cein(M)$. The paper contains several open questions.

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arXiv2014

Remarks on Bianchi sums and Pontrjagin classes

We use the exterior and composition products of double forms together with the alternating operator to reformulate Pontrjagin classes and all Pontrjagin numbers in terms of the Riemannian curvature. We show that the alternating operator is obtained by a succession of applications of the first Bianchi sum and we prove some useful identities relating the previous four operations on double forms. As an application, we prove that for a $k$-conformally flat manifold of dimension $n\geq 4k$, the Pontrjagin classes $P_i$ vanish for any $i\geq k$. Finally, we study the equality case in an inequality of Thorpe between the Euler-Poincaré charateristic and the $k$-th Pontrjagin number of a $4k$-dimensional Thorpe manifold.

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arXiv2013

On some algebraic identities and the exterior product of double forms

We use the exterior product of double forms to reformulate celebrated classical results of linear algebra about matrices and bilinear forms namely the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton identities and Jacobi's formula for the determinant. This new formalism is then used to naturally generalize the previous results to higher multilinear forms namely to double forms. In particular, we show that the Cayley-Hamilton theorem once applied to the second fundamental form of a hypersurface of the Euclidean space is equivalent to a linearized version of the Gauss-Bonnet theorem, and once its generalization is applied to the Riemann curvature tensor (seen as a $(2,2)$ double form) is an infinitisimal version of the general Gauss-Bonnet-Chern theorem. In addition to that, the general Cayley-Hamilton theorems generate several universal curvature identities. The generalization of the classical Laplace expansion of the determinant to double forms is shown to lead to new general Avez type formulas for all Gauss-Bonnet curvatures.

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arXiv2011

About Some Quadratic Scalar Curvatures and the $h_{4}$ Yamabe Equation

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 at leat 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.

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arXiv2008

On Generalized Einstein Metrics

Recall that 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 Thorpe tensors, one gets several natural generalizations of Einstein's condition. In this paper, we study some properties of these classes of metrics.

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arXiv2007

On Gauss-Bonnet Curvatures

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.

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arXiv2006

Riemannian curvature: variations on different notions of positivity

We study different notions of Riemannian curvatures: The $p$-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the $(p,q)$-curvatures, which incorporate all the previous curvatures. We then examine the curvature term which appears in the classical Weitzenböck formula. We also study the positivity properties of the $p$-curvatures, the second Gauss-Bonnet-Weyl curvature, the Einstein curvature and the isotropic curvature.

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arXiv2006

On Weitzenböck Curvature Operators

The Weitzenböck curvature operators are the curvature terms of order zero that appear in the well known classical Weitzenböck formula. In this paper, we use the formalism of double forms to prove a simple formula for this operators and to study their geometric properties.

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