arXiv research

Two reduction schemes for symplectic manifolds are shown equivalent.

We show that the contact reduction can be specialized to Sasakian manifolds. We link this Sasakian reduction to Kähler reduction by considering the Kähler cone over a Sasakian manifold. We present examples of Sasakian manifolds obtained by $S^1$ reduction of standard Sasakian spheres.

We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invari…

Study characterizes naturally reductive metrics on homogeneous manifolds.

We define reduction of locally conformal Kaehler manifolds, considered as conformal Hermitian manifolds, and we show its equivalence with an unpublished construction given by Biquard and Gauduchon. We show the compatibility between this reduction and Kaehler reduction of the universal cover. By a recent result of Kamis…

This work introduces a unified approach to the reduction of Poisson manifolds using their description by graded symplectic manifolds. This yields a generalization of the classical Poisson reduction by distributions (Marsden-Ratiu reduction). Further it allows one to construct actions of strict Lie 2-groups and to descr…

Study star products on Poisson manifolds compatible with reduction.

Reduces symplectic manifolds with singularities for quantum reduction.

We study reduction of generalized complex structures. More precisely, we investigate the following question. Let $J$ be a generalized complex structure on a manifold $M$, which admits an action of a Lie group $G$ preserving $J$. Assume that $M_0$ is a $G$-invariant smooth submanifold and the $G$-action on $M_0$ is prop…

Survey on geometric foundations of data reduction methods.

The paper simplifies symmetries in complex geometric structures.

Paper shows spectra can't distinguish naturally reductive manifolds.

Reduces Lie (bi-)algebroids and Dirac manifolds using constraint vector bundles.

The version of Marsden-Ratiu reduction theorem for Nambu-Poisson manifolds by a regular distribution has been studied by Ib$\acute{\text{a}}\tilde{\text{n}}$ez et al. In this paper we show that the reduction is always ensured unless the distribution is zero. Next we extend the more general Falceto-Zambon Poisson reduct…

In this paper, we develop results in the direction of an analogue of Sjamaar and Lerman's singular reduction of Hamiltonian symplectic manifolds in the context of reduction of Hamiltonian generalized complex manifolds (in the sense of Lin and Tolman). Specifically, we prove that if a compact Lie group acts on a general…

This paper classifies instantons with closed reductions and provides examples of non-closed reductions.

Paper proves naturally reductive property is inaudible for certain manifolds.

This report concerns the problem of dimensionality reduction through information geometric methods on statistical manifolds. While there has been considerable work recently presented regarding dimensionality reduction for the purposes of learning tasks such as classification, clustering, and visualization, these method…

Contact reductions explained through symplectic reductions.

In this note we give conditions which ensure the reduction of a symplectic connection in the process of a Marsden-Weinstein reduction and of the reduction of a presymplectic manifold.

Study of symplectic and Poisson reduction, proposing Poisson implosion.

Reduces observables on multisymplectic manifolds using Lie algebra actions.

Study of symplectic trivialization and reduction of bundles with symmetry and connection.

We classify totally geodesic and parallel hypersurfaces of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds.

Reduces Poisson manifolds with Hamiltonian Lie algebroids.

A method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If…

Study homogeneous Lorentzian manifolds under reductive Lie groups, reducing descriptions to semisimple groups.

We prove a reduction theorem for the tangent bundle of a Poisson manifold $(M, π)$ endowed with a pre-Hamiltonian action of a Poisson Lie group $(G, π_G)$. In the special case of a Hamiltonian action of a Lie group, we are able to compare our reduction to the classical Marsden-Ratiu reduction of $M$. If the manifold $M…

New definition of naturally reductive Finsler manifolds using geodesic graphs.

We introduce and study the notion of Sasaki--Weyl manifold, which is a natural generalization of the notion of Sasaki manifold. We construct a reduction of Sasaki--Weyl manifolds and we show that it commutes with several reductions already existing in the literature.

It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new method to construct invariant non-naturally reductive Einstein metrics on normal ho…

The abstract discusses conditions for hyperkähler manifolds and Kähler reduction.

We reframe linear dimensionality reduction as a problem of Bayesian inference on matrix manifolds. This natural paradigm extends the Bayesian framework to dimensionality reduction tasks in higher dimensions with simpler models at greater speeds. Here an orthogonal basis is treated as a single point on a manifold and is…

This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases, this theory includes not only Lagrangian reduction (including reduction by stages) for Lie group actions, but also classica…

Every holomorphic effective parabolic or reductive geometry on a domain over a Stein manifold extends uniquely to the envelope of holomorphy of the domain. This result completes the open problems of my earlier paper on extension of holomorphic geometric structures on complex manifolds. We use this result to classify th…

A new framework learns clustering and dimensionality reduction together.

A new method for reducing model complexity using neural active manifolds.

Given the Euclidean space $\R^{2n+2}$ endowed with a constant symplectic structure and the standard flat connection, and given a polynomial of degree 2 on that space, Baguis and Cahen have defined a reduction procedure which yields a symplectic manifold endowed with a Ricci-type connection. We observe that any symplect…

We extend the Falceto-Zambon version of Marsden-Ratiu Poisson reduction to Poisson quasi-Nijenhuis structures with background on manifolds. We define gauge transformations of Poisson quasi-Nijenhuis structures with background, study some of their properties and show that they are compatible with reduction procedure. We…

We classify non-reductive four-dimensional homogeneous conformally Einstein manifolds.

Study of hyperkähler reduction on abelian varieties and toric manifolds.

Quantization and reduction studied for CR manifolds with group actions.

We obtain universal models for several types of locally conformal symplectic manifolds via pullback or reduction. The relation with recent embedding results for locally conformal Kähler manifolds is discussed.

We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field.

Study bihamiltonian structures and Frobenius manifolds for specific Toda hierarchies.

Study surjectivity of Kirwan map for generalized hyperkähler reduction.

Study on instantons over product manifolds with a codimension-4 form.

Classifies compact multiplicity free quasi-Hamiltonian manifolds.