arXiv subjects

Brice Flamencourt

Publications and source records attributed to Brice Flamencourt.

13 recordsLinked to original sources

arXiv2026

Arithmetic structure of generalized Inoue--Bombieri manifolds

A Generalized Inoue--Bombieri (GIB) manifold $M$ is a compact quotient of a connected Riemannian product $\mathbb{R}^q \times (N,g _N)$ by a discrete subgroup of $\mathrm{Sim}(\mathbb{R}^q) \times \mathrm{Isom}(N,g_N)$. The flat factor induces a transversely Riemannian foliation whose leaf closures determine, up to a natural geometric modification, a torus fibration $M \to X$. The main goal of this article is to study the associated monodromy representation $ρ: π_1(X) \to \mathrm{GL}(n,\mathbb{Z})$. We prove that the image of $ρ$ is a subgroup of a cocompact arithmetic lattice of a reductive group, and we discuss which groups may be realized as monodromy groups of GIB manifolds. When $(N,g_N)$ is a symmetric space of non-compact type, the monodromy itself is arithmetic. Moreover, one may describe the fibration and the monodromy in terms of parabolic subgroups of the isometry group of $(N,g_N)$. This yields new examples of GIB manifolds, as well as obstructions, and opens the way toward a complete classification in this particular case.

math.DG↗

arXiv2025

On the Inoue-Bombieri construction

We study compact quotients of a Riemannian product $\mathbb{R}^q \times (N, g_N)$, where $(N, g_N)$ is a complete Riemannian manifold, by discrete subgroups $Γ$ of $\mathrm{Sim}(\mathbb{R}^q) \times \mathrm{Isom}(N)$. When $N$ is a symmetric space of non-compact type, this construction generalizes the well-known Inoue--Bombieri surfaces. We show that this setting is actually equivalent to that of the so-called LCP manifolds, and we establish a Bieberbach-type rigidity result in the case where $N$ is symmetric. In addition, we provide a classification of the manifolds $N$ and the groups $Γ$ when $N$ is a Hadamard manifold with strictly negative curvature.

math.DG↗

arXiv2025

On foliations admitting a transverse similarity structure

We give a "conceptual" approach to Kourganoff's results about foliations with a transverse similarity structure. In particular, we give a proof, understandable by the targeted community, of the very important result classifying the holonomy of the closed, non-exact Weyl structures on compact manifolds, from which arose the notion of locally conformally product structures. We also extract from the proof several results on foliations admitting locally metric transverse connections.

math.DG↗

arXiv2025

Weyl structures with special holonomy on compact conformal manifolds

We consider compact conformal manifolds $(M,[g])$ endowed with a closed Weyl structure $\nabla$, i.e. a torsion-free connection preserving the conformal structure, which is locally but not globally the Levi-Civita connection of a metric in $[g]$. Our aim is to classify all such structures when both $\nabla$ and $\nabla^g$, the Levi-Civita connection of $g$, have special holonomy. In such a setting, $(M,[g],\nabla)$ is either flat, or irreducible, or carries a locally conformally product (LCP) structure. Since the flat case is already completely classified, we focus on the last two cases. When $\nabla$ has irreducible holonomy we prove that $(M,g)$ is either Vaisman, or a mapping torus of an isometry of a compact nearly Kähler or nearly parallel $\mathrm{G}_2$ manifold, while in the LCP case we prove that $g$ is neither Kähler nor Einstein, thus reducible by the Berger-Simons Theorem, and we obtain the local classification of such structures in terms of adapted metrics.

math.DG↗

arXiv2025

Flat degenerate metrics and Riemannian foliations

Bandyopadhyay, Dacorogna, Matveev and Troyanov conjectured that a closed manifold admitting a flat, non-negative definite metric of constant rank $m$ should be finitely covered by a fiber bundle over the $m$-torus. We give a counter-example to this statement and we discuss the link between this problem and the study of transversely flat Riemannian foliations.

math.DG↗

arXiv2024

Conformal vector fields on LCP manifolds

We show that conformal vector fields on compact locally conformally product manifolds are orthogonal to the flat distribution and Killing with respect to the Gauduchon metric.

math.DG↗

arXiv2024

Torsion-free connections on $G$-structures

We prove that for a group $\mathrm{SO}_n(\mathrm{R}) \subset G \subset \mathrm{GL}_n (\mathrm{R})$, any $G$-structure on a smooth manifold can be endowed with a torsion free connection which is locally the Levi-Civita connection of a Riemannian metric in a given conformal class. In this process, we classify the admissible groups.

math.DG↗

arXiv2024

Locally conformally product structures

A locally conformally product (LCP) structure on compact manifold $M$ is a conformal structure $c$ together with a closed, non-exact and non-flat Weyl connection $D$ with reducible holonomy. Equivalently, an LCP structure on $M$ is defined by a reducible, non-flat, incomplete Riemannian metric $h_D$ on the universal cover $\tilde M$ of $M$, with respect to which the fundamental group $π_1(M)$ acts by similarities. It was recently proved by Kourganoff that in this case $(\tilde M, h_D)$ is isometric to the Riemannian product of the flat space $\mathbb{R}^q$ and an incomplete irreducible Riemannian manifold $(N,g_N)$. In this paper we show that for every LCP manifold $(M,c,D)$, there exists a metric $g\in c$ such that the Lee form of $D$ with respect to $g$ vanishes on vectors tangent to the distribution on $M$ defined by the flat factor $\mathbb{R}^q$, and use this fact in order to construct new LCP structures from a given one by taking products. We also establish links between LCP manifolds and number field theory, and use them in order to construct large classes of examples, containing all previously known examples of LCP manifolds constructed by Matveev-Nikolayevsky, Kourganoff and Oeljeklaus-Toma.

math.DG↗

arXiv2024

On decomposable LCP structures

We introduce the notion of decomposable locally conformally product (LCP) manifolds and characterize those which are defined on quotients of Riemannian Lie groups by co-compact lattices.

math.DG↗

arXiv2024

The characteristic group of locally conformally product structures

A compact manifold $M$ together with a Riemannian metric $h$ on its universal cover $\tilde M$ for which $π_1(M)$ acts by similarities is called a similarity structure. In the case where $π_1(M) \not\subset \mathrm{Isom}(\tilde M, h)$ and $(\tilde M, h)$ is reducible but not flat, this is a Locally Conformally Product (LCP) structure. The so-called characteristic group of these manifolds, which is a connected abelian Lie group, is the key to understand how they are built. We focus in this paper on the case where this group is simply connected, and give a description of the corresponding LCP structures. It appears that they are quotients of trivial $\mathbb{R}^p$-principal bundle over simply-connected manifolds by certain discrete subgroups of automorphisms. We prove that, conversely, it is always possible to endow such quotients with an LCP structure.

math.DG↗

arXiv2021

Cauchy spinors on $3$-manifolds

Let $\mathcal{Z}$ be a spin $4$-manifold carrying a parallel spinor and $M\hookrightarrow \mathcal{Z}$ a hypersurface. The second fundamental form of the embedding induces a flat metric connection on $TM$. Such flat connections satisfy a non-elliptic, non-linear equation in terms of a symmetric $2$-tensor on $M$. When $M$ is compact and has positive scalar curvature, the linearized equation has finite dimensional kernel. Four families of solutions are known on the round $3$-sphere $\mathbb{S}^3$. We study the linearized equation in the vicinity of these solutions and we construct as a byproduct an incomplete hyperkähler metric on $\mathbb{S}^3\times \mathbb{R}$ closely related to the Euclidean Taub-NUT metric on $\mathbb{R}^4$. On $\mathbb{S}^3$ there do not exist other solutions which either are constant in a left (or right) invariant frame, have three distinct constant eigenvalues, or are invariant in the direction of a left (or right)-invariant eigenvector. We deduce from this last result an extension of Liebmann's sphere rigidity theorem.

math.DG↗

arXiv2021

A generalized MIT Bag operator on spin manifolds in the non-relativistic limit

We consider Dirac-like operators with piecewise constant mass terms on spin manifolds, and we study the behaviour of their spectra when the mass parameters become large. In several asymptotic regimes, effective operators appear: the extrinsic Dirac operator and a generalized MIT Bag Dirac operator. This extends some results previously known for the Euclidean spaces to the case of general spin geometry.

math.SP↗

arXiv2019

Strong coupling asymptotics for $δ$-interactions supported by curves with cusps

Let $Γ\subset \mathbb{R}^2$ be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve $|x_2|=x_1^p$ for some $p>1$. We study the eigenvalues of the Schrödinger operator $H_α$ with the attractive $δ$-potential of strength $α>0$ supported by $Γ$, which is defined by its quadratic form \[ H^1(\mathbb{R}^2)\ni u\mapsto \iint_{\mathbb{R}^2} |\nabla u|^2\,\mathrm{d}x-α\int_Γu^2\, \mathrm{d}s, \] where $\mathrm{d}s$ stands for the one-dimensional Hausdorff measure on $Γ$. It is shown that if $n\in\mathbb{N}$ is fixed and $α$ is large, then the well-defined $n$th eigenvalue $E_n(H_α)$ of $H_α$ behaves as \[ E_n(H_α)=-α^2 + 2^{\frac{2}{p+2}} \mathcal{E}_n \,α^{\frac{6}{p+2}} + \mathcal{O}(α^{\frac{6}{p+2}-η}), \] where the constants $\mathcal{E}_n>0$ are the eigenvalues of an explicitly given one-dimensional Schrödinger operator determined by the cusp, and $η>0$. Both main and secondary terms in this asymptotic expansion are different from what was observed previously for the cases when~$Γ$ is smooth or piecewise smooth with non-zero angles.

math.SP↗