arXiv subjects

Eckhard Meinrenken

Publications and source records attributed to Eckhard Meinrenken.

At least 19 recordsLinked to original sources

arXiv2026

A note on the cohomology of pullback Lie algebroids

We give a direct differential-geometric proof of a result of Crainic on the cohomology of pullbacks of Lie algebroids under surjective submersions: If the fibers are connected and have vanishing cohomology through degree $n$, pullback with coefficients in any representation is an isomorphism through degree $n$ and is injective in degree $n+1$. The proof reduces to a vanishing result for leafwise cohomology and uses an exhaustion-and-patching argument due to Buchdahl. No local triviality or finite-type assumption on the fibers is required. We also prove the analogous result for holomorphic Lie algebroids.

math.DG↗

arXiv2026

An introduction to weightings along submanifolds

This article is based on a talk given at the Ghent Geometric Analysis Seminar in 2023. We review basic notions from the theory of weightings along submanifolds, with special emphasis on multiplicative weightings for Lie groupoids along subgroupoids.

math.DG↗

arXiv2026

Symplectic geometry of projective structures on surfaces with boundary

For oriented surfaces $Σ$ with boundary, we consider the infinite-dimensional deformation space of projective structures on $Σ$ with nondegenerate boundary, up to isotopies fixing the boundary. We show that this space carries a natural symplectic structure, and is a Hamiltonian space for the symplectic groupoid integrating the Adler-Gelfand-Dikii-space of the boundary.

math.SG↗

arXiv2025

Introduction to moduli spaces and Dirac geometry

Let $G$ be a Lie group, with an invariant metric on its Lie algebra $\mathfrak{g}$. Given a surface $Σ$ with boundary, and a collection of base points $\mathcal{V}\subset Σ$ meeting every boundary component, the moduli space (representation variety) $\mathcal{M}_G(Σ,\mathcal{V})$ carries a distinguished `quasi-symplectic' 2-form. We shall explain the finite-dimensional construction of this 2-form and discuss its basic properties, using quasi-Hamiltonian techniques and Dirac geometry. This article is an extended version of lectures given at the summer school 'Poisson 2024' at the Accademia Pontaniana in Napoli, July 2024.

math.DG↗

arXiv2025

Manin pairs and moment maps revisited

The notion of quasi-Poisson $G$-spaces with $D/G$-valued moment maps was introduced by Alekseev and Kosmann-Schwarzbach in 1999. Our main result is a \emph{Lifting Theorem}, establishing a bijective correspondence between the categories of quasi-Poisson $G$-spaces with $D/G$-valued moment maps and of quasi-Poisson $G\times G$-spaces with $D$-valued moment maps. Using this result, we give simple constructions of fusion and conjugation for these spaces, and new examples coming from moduli spaces.

math.DG↗

arXiv2024

On the integration of Manin pairs

It is a remarkable fact that the integrability of a Poisson manifold to a symplectic groupoid depends only on the integrability of its cotangent Lie algebroid $A$: The source-simply connected Lie groupoid $G\rightrightarrows M$ integrating $A$ automatically acquires a multiplicative symplectic 2-form. More generally, a similar result holds for the integration of Lie bialgebroids to Poisson groupoids, as well as in the `quasi' settings of Dirac structures and quasi-Lie bialgebroids. In this article, we will place these results into a general context of Manin pairs $(\mathbb{E},A)$, thereby obtaining a simple geometric approach to these integration results. We also clarify the case where the groupoid $G$ integrating $A$ is not source-simply connected. Furthermore, we obtain a description of Hamiltonian spaces for Poisson groupoids and quasi-symplectic groupoids within this formalism.

math.DG↗

arXiv2024

Lie Algebroids

This is an overview article on Lie algebroids, and their role as the infinitesimal counterparts of Lie groupoids.

math.DG↗

arXiv2024

Symplectic geometry of Teichmüller spaces for surfaces with ideal boundary

A hyperbolic 0-metric on a surface with boundary is a hyperbolic metric on its interior, exhibiting the boundary behavior of the standard metric on the Poincaré disk. Consider the infinite-dimensional Teichmüller spaces of hyperbolic 0-metrics on oriented surfaces with boundary, up to diffeomorphisms fixing the boundary and homotopic to the identity. We show that these spaces have natural symplectic structures, depending only on the choice of an invariant metric on sl(2,R). We prove that these Teichmüller spaces are Hamiltonian Virasoro spaces for the action of the universal cover of the group of diffeomorphisms of the boundary. We give an explicit formula for the Hill potential on the boundary defining the moment map. Furthermore, using Fenchel-Nielsen parameters we prove a Wolpert formula for the symplectic form, leading to global Darboux coordinates on the Teichmüller space.

math.DG↗

arXiv2022

On the coadjoint Virasoro action

The set of coadjoint orbits of the Virasoro algebra at level 1 is in bijection with the set of conjugacy classes in a certain open subset $\widetilde{\rm SL}(2,\mathbb{R})_+$ of the universal cover of ${\rm SL}(2,\mathbb{R})$. We strengthen this bijection to a Morita equivalence of quasi-symplectic groupoids, integrating the Poisson structure on $\mathfrak{vir}^*_\mathsf{1}(S^1)$ and the Cartan-Dirac structure on $\widetilde{\rm SL}(2,\mathbb{R})_+$, respectively.

math.DG↗

arXiv2022

Singular Lie filtrations and weightings

We study weightings (a.k.a. quasi-homogeneous structures) arising from manifolds with singular Lie filtrations. This generalizes constructions of Choi-Ponge, Van Erp-Yuncken, and Haj-Higson for (regular) Lie filtrations.

math.DG↗

arXiv2021

Differential Geometry of Weightings

We describe the notion of a \emph{weighting} along a submanifold $N\subset M$, and explore its differential-geometric implications. This includes a detailed discussion of weighted normal bundles, weighted deformation spaces, and weighted blow-ups. We give a description of weightings in terms of subbundles of higher tangent bundles, which leads us to notions of multiplicative weightings for Lie algebroids and Lie groupoids.

math.DG↗

arXiv2021

Quotients of double vector bundles and multigraded bundles

We study quotients of multi-graded bundles, including double vector bundles. Among other things, we show that any such quotient fits into a tower of affine bundles. Applications of the theory include a construction of normal bundles for weighted submanifolds, as well as for pairs of submanifolds with clean intersection.

math.DG↗

arXiv2021

On the integration of transitive Lie algebroids

We revisit the problem of integrating Lie algebroids $A\Rightarrow M$ to Lie groupoids $G\rightrightarrows M$, for the special case that the Lie algebroid $A$ is transitive. We obtain a geometric explanation of the Crainic-Fernandes obstructions for this situation, and an explicit construction of the integration whenever these obstructions vanish. We also indicate an extension of this approach to regular Lie algebroids.

math.DG↗

arXiv2020

Euler-like vector fields, normal forms, and isotropic embeddings

Germs of tubular neighborhood embeddings for submanifolds N of manifolds M are in one-one correspondence with germs of Euler-like vector fields near N. In many contexts, this reduces the proof of `normal forms results' for geometric structures to the construction of an Euler-like vector field compatible with the given structure. We illustrate this principle in a variety of examples, including the Morse-Bott lemma, Weinstein's Lagrangian embedding theorem, and Zung's linearization theorem for proper Lie groupoids. In the second part of this article, we extend the theory to a weighted context, with an application to isotropic embeddings.

math.DG↗

arXiv2019

The Weil algebra of a double Lie algebroid

Given a double vector bundle $D\to M$, we define a bigraded `Weil algebra' $\mathcal{W}(D)$, which `realizes' the algebra of smooth functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebras of $D$ and those of the double vector bundles $D',\ D"$ obtained by duality operations. In particular, we show that double-linear Poisson structures on $D$ can be described alternatively as Gerstenhaber brackets on $\mathcal{W}(D)$, vertical differentials on $\mathcal{W}(D')$, or horizontal differentials on $\mathcal{W}(D")$. We also give a new proof of Voronov's result characterizing double Lie algebroid structures. In the case that $D=TA$ is the tangent prolongation of a Lie algebroid, we find that $\mathcal{W}(D)$ is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad-Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multivector fields, and 2-term representations up to homotopy all have natural interpretations in terms of our Weil algebras.

math.DG↗

arXiv2019

Van Est differentiation and integration

The classical Van Est theory relates the smooth cohomology of Lie groups with the cohomology of the associated Lie algebra, or its relative versions. Some aspects of this theory generalize to Lie groupoids and their Lie algebroids. In this paper, continuing an idea from [18], we revisit the van Est theory using the Perturbation Lemma from homological algebra. Using this technique, we obtain precise results for the van Est differentiation and integrations maps at the level of cochains. Specifically, we construct homotopy inverses to the van Est differentiation maps that are right inverses at the cochain level.

math.DG↗

arXiv2018

Deformation spaces and normal forms around transversals

Given a manifold M with a submanifold N, the deformation space D(M,N) is a manifold with a submersion to R whose zero fiber is the normal bundle, and all other fibers are equal to M. This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, with N a submanifold transverse to the foliation. New examples include L_\infty-algebroids, Courant algebroids, and Lie bialgebroids. In each case, we obtain a normal form theorem around N, in terms of a model structure over the normal bundle.

math.DG↗

arXiv2018

The decomposition formula for Verlinde Sums

We prove a decomposition formula for Verlinde sums (rational trigonometric sums), as a discrete counterpart to the Boysal-Vergne decomposition formula for Bernoulli series. Motivated by applications to fixed point formulas in Hamiltonian geometry, we develop differential form valued version of Bernoulli series and Verlinde sums, and extend the decomposition formula to this wider context.

math.SG↗