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Madeleine Jotz

Publications and source records attributed to Madeleine Jotz.

12 recordsLinked to original sources
arXiv2023

A geometrisation of $\mathbb N$-manifolds

This paper proposes a geometrisation of $\mathbb N$-manifolds of degree $n$ as $n$-fold vector bundles equipped with a (signed) $S_n$-symmetry. More precisely, it proves an equivalence between the categories of $[n]$-manifolds and the category of symmetric $n$-fold vector bundles, by finding that symmetric $n$-fold vector bundle cocycles and $[n]$-manifold cocycles are identical. This extends the already known equivalences of $[1]$-manifolds with vector bundles, and of $[2]$-manifolds with involutive double vector bundles, where the involution is understood as an $S_2$-action.

math.DG↗
arXiv2012

Glanon groupoids

We introduce the notion of Glanon groupoids, which are Lie groupoids equipped with multiplicative generalized complex structures. It combines symplectic groupoids, holomorphic Lie groupoids and holomorphic Poisson groupoids into a unified framework. Their infinitesimal, Glanon Lie algebroids are studied. We prove that there is a bijection between Glanon Lie algebroids and source-simply connected and source-connected Glanon groupoids. As a consequence, we recover various integration theorem and obtain the integration theorem for holomorphic Poisson groupoids.

math.DG↗
arXiv2012

Foliated groupoids and their infinitesimal data

In this work, we study Lie groupoids equipped with multiplicative foliations and the corresponding infinitesimal data. We determine the infinitesimal counterpart of a multiplicative foliation in terms of its core and sides together with a partial connection satisfying special properties, giving rise to the concept of IM-foliation on a Lie algebroid. The main result of this paper shows that if $G$ is a source simply connected Lie groupoid with Lie algebroid $A$, then there exists a one-to-one correspondence between multiplicative foliations on $G$ and IM-foliations on the Lie algebroid $A$.

math.DG↗
arXiv2011

Singular reduction of Dirac structures

The regular reduction of a Dirac manifold acted upon freely and properly by a Lie group is generalized to a nonfree action. For this, several facts about $G$-invariant vector fields and one-forms are shown.

math.DG↗
arXiv2011

Induced Dirac structure on isotropy type manifolds

A new method of singular reduction is extended from Poisson to Dirac manifolds. Then it is shown that the Dirac structures on the strata of the quotient coincide with those of the only other known singular Dirac reduction method.

math.DG↗
arXiv2011

The leaf space of a multiplicative foliation

We show that if a smooth multiplicative subbundle $S\subseteq TG$ on a groupoid $G\rr P$ is involutive and satisfies completeness conditions, then its leaf space $G/S$ inherits a groupoid structure over the space of leaves of $TP\cap S$ in $P$. As an application, a special class of Dirac groupoids is shown to project by forward Dirac maps to Poisson groupoids.

math.DG↗
arXiv2011

Dirac optimal reduction

The purpose of this paper is to generalize the regular Optimal Reduction Theorem to general proper Dirac actions, formulated both in terms of point and orbit reduction. A comparison to general standard singular Dirac reduction is given emphasizing the desingularization role played by optimal reduction.

math.DG↗
arXiv2011

Dirac Lie groups, Dirac homogeneous spaces and the Theorem of Drinfeld

The notions of \emph{Poisson Lie group} and \emph{Poisson homogeneous space} are extended to the Dirac category. The theorem of Drinfel$'$d (\cite{Drinfeld93}) on the one-to-one correspondence between Poisson homogeneous spaces of a Poisson Lie group and a special class of Lagrangian subalgebras of the Lie bialgebra associated to the Poisson Lie group is proved to hold in this more general setting.

math.DG↗
arXiv2009

Hedlund-Metrics and the Stable Norm

The real homology of a compact Riemannian manifold $M$ is naturally endowed with the stable norm. The stable norm on $H_1(M,\mathbb{R})$ arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finite-dimensional vector space $H_1(M,\mathbb{R})$ are stable norms of a Riemannian metric on $M$. If the dimension of $M$ is at least three, I. Babenko and F. Balacheff proved in \cite{baba} that every polyhedral norm ball in $H_1(M,\mathbb{R})$, whose vertices are rational with respect to the lattice of integer classes in $H_1(M,\mathbb{R})$, is the stable norm ball of a Riemannian metric on $M$. This metric can even be chosen to be conformally equivalent to any given metric. The proof in \cite{baba} uses singular Riemannian metrics on polyhedra which are finally smoothed. Here we present an alternative construction of such metrics which remains in the geometric framework of smooth Riemannian metrics.

math.DG↗
arXiv2008

Closedness of the tangent spaces to the orbits of proper actions

In this note we show that for any proper action of a Banach--Lie group $G$ on a Banach manifold $M$, the corresponding tangent maps $\g \to T_x(M)$ have closed range for each $x \in M$, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $M/G$ carries a natural manifold structure.

math.DG↗