A new method for classifying naturally reductive spaces is presented. This method relies on the structure theory of naturally reductive spaces developed in \cite{Storm2018a} and the new construction of naturally reductive spaces in \cite{Storm2018}. We obtain the classification of all naturally reductive spaces in dime…
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A new construction of naturally reductive spaces is presented. This construction gives a large amount of new families of naturally reductive spaces. First the infinitesimal models of the new naturally reductive spaces are constructed. A concrete transitive group of isometries is given for the new spaces and also the na…
In the present paper we study naturally reductive homogeneous -metric spaces. Under some conditions, we give some necessary and sufficient conditions for a homogeneous -metric space to be naturally reductive. Then we show that for such spaces the two definitions of naturally reductive homogeneous Finsler …
Study extends Kobayashi's method to non-reductive subgroups for homogeneous spaces.
problem Existence of compact Clifford-Klein forms in homogeneous spaces.
method Extend Kobayashi's method to non-reductive subgroups and compare Cartan projections and non-compact dimensions.
result Examples of homogeneous spaces without compact Clifford-Klein forms.
New findings on Codazzi tensors in homogeneous spaces.
problem Characterizing Codazzi tensor fields in reductive homogeneous spaces.
method Extending results from Lie groups to reductive homogeneous spaces, analyzing the curvature of canonical connections.
result Invariant Codazzi tensor fields on naturally reductive homogeneous spaces are parallel.
New method reduces version space for CNNs, improving active learning performance.
problem Sampling bias in active learning hinders optimal hypothesis finding in neural networks.
method Version space reduction through prior mass reduction and diameter reduction, proposing a new Gibbs-vote disagreement method.
result Diameter-based querying method reduces version space more effectively than prior mass reduction and other methods.
The main result of this paper is that every naturally reductive space can be explicitly constructed from the construction in \cite{Storm2018}. This gives us a general formula for any naturally reductive space and from this we prove reducibility and isomorphism criteria.
We consider an enlarged dimension reduction space in functional inverse regression. Our operator and functional analysis based approach facilitates a compact and rigorous formulation of the functional inverse regression problem. It also enables us to expand the possible space where the dimension reduction functions bel…
A family of naturally reductive pseudo-Riemannian spaces is constructed out of the representations of Lie algebras with ad-invariant metrics. We exhibit peculiar examples, study their geometry and characterize the corresponding naturally reductive homogeneous structure.
A new algebraic structure emerges from reductive homogeneous spaces.
problem Understanding the algebraic properties of tangent bundles.
method Defined a new algebraic structure based on connections and torsion.
result Post-Lie-Yamaguti algebra is a new algebraic structure related to Lie-Yamaguti algebras.
The classification of 4-dimensional naturally reductive pseudo-Riemannian spaces is given. This classification comprises symmetric spaces, the product of 3-dimensional naturally reductive spaces with the real line and new families of indecomposable manifolds which are studied at the end of the article. The oscillator g…
Parallel transport map over reductive spaces is an affine submersion.
problem Understanding parallel transport in reductive homogeneous spaces with torsion.
method Generalizing previous results on affine symmetric spaces, proving compactness of shape operators, and proposing definitions for regularized mean curvatures.
result Each fiber of the parallel transport map over a reductive homogeneous space is minimal in both senses.
Naturally reductive spaces, in general, can be seen as an adequate generalization of Riemannian symmetric spaces. Nevertheless, there are some that are closer to symmetric spaces than others. On the one hand, there is the series of Hopf fibrations over complex space forms, including the Heisenberg groups with their met…
The paper introduces controllable principal connections and estimates distances between bundles and spaces.
problem Estimating distances between bundles and spaces using controllable connections.
method Combining orbit theorem, Ambrose-Singer theorem, and controllable principal connections.
result Proves convergence of metrics to normal reductive homogeneous spaces.
Introduces a new method for symplectic reduction along submanifolds.
problem Symplectic reduction in various geometric categories.
method Uniform approach to symplectic reduction in smooth manifolds, complex analytic spaces, and algebraic varieties.
result Generalizes and encompasses various known reduction techniques.
We study invariant metrics on Ledger-Obata spaces . We give the classification and an explicit construction of all naturally reductive metrics, and also show that in the case , any invariant metric is naturally reductive. We prove that a Ledger-Obata space is a geodesic orbit space if and onl…
In this paper we give a short geometric proof of a generalization of a well-known result about reduction of codimension for submanifolds of Riemannian symmetric spaces.
Study flag curvature in homogeneous Finsler spaces with a specific metric.
problem Analyzing flag curvature in homogeneous Finsler spaces with a generalized -Kropina metric.
method Provided explicit formula for flag curvature, showed equivalence of definitions, and studied curvature of naturally reductive spaces.
result Equivalence of two definitions of naturally reductive homogeneous Finsler spaces for the generalized -Kropina metric.
The main purpose of the following article is to introduce a \emph{Lie theoretical} approach to the problem of classifying pseudo quaternionic-Kähler (QK) reductions of the pseudo QK symmetric spaces, otherwise called \emph{generalized Wolf spaces}.
Reduces observables on multisymplectic manifolds using Lie algebra actions.
problem Reduction of observables on multisymplectic manifolds with Lie algebra actions.
method Development of a reduction scheme for -algebra of observables.
result Reproduces symplectic observable reduction in specific cases.
Geodesic orbit metrics on homogeneous spaces constructed by strongly isotropy irreducible spacesmath.DG
Geodesic orbit metrics proven on specific homogeneous spaces.
problem Geodesic orbit metrics on homogeneous spaces.
method Using strongly isotropy irreducible spaces and proving natural reductivity.
result Geodesic orbit metrics are naturally reductive on constructed homogeneous spaces.
Movement primitives are an important policy class for real-world robotics. However, the high dimensionality of their parametrization makes the policy optimization expensive both in terms of samples and computation. Enabling an efficient representation of movement primitives facilitates the application of machine learni…
The purpose of these survey notes is to give a presentation of a classical theorem of Nomizu that relates the invariant affine connections on reductive homogeneous spaces and nonassociative algebras.
We classify non-reductive four-dimensional homogeneous conformally Einstein manifolds.
Four-dimensional compact Clifford-Klein forms of pseudo-Riemannian symmetric spaces with signature math.DG
Classifies 4D spaces with compact Clifford-Klein forms.
problem Classifying 4D symmetric spaces with compact Clifford-Klein forms.
method Developed a method for 1-connected solvable symmetric spaces.
result Classification of 4D spaces with compact Clifford-Klein forms.
Paper finds surface groups can deform in reductive symmetric spaces.
problem Finding deformations of discontinuous groups in reductive symmetric spaces.
method Analyzing Zariski dense surface subgroups and their deformations.
result Surface groups of high genus can deform in reductive symmetric spaces.
We introduce a novel data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space wh…
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…
Rollings of reductive homogeneous spaces are studied using intrinsic curves.
problem Investigate rollings of reductive homogeneous spaces without slip and twist.
method An intrinsic point of view, considering rollings as curves in the configuration space tangent to a certain distribution.
result Explicit solutions for rollings of over are obtained for specific cases.
A method constructs a stochastic surrogate from dimensionality reduction results for high-dimensional uncertainty quantification.
problem High-dimensional uncertainty quantification with physics-based models.
method Constructs a stochastic surrogate model from dimensionality reduction results.
result Preserves convenience of sequential dimensionality reduction and Gaussian process regression while overcoming limitations.
We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which is tangent o…
Reduces symplectic manifolds with singularities for quantum reduction.
problem Quantization commutes with reduction for singular symplectic manifolds.
method Reduction theory for -symplectic manifolds and folded symplectic manifolds under general symmetries.
result New constructions of (singular) quasi-Hamiltonian spaces via reduction and fusion product.
Study on instantons over product manifolds with a codimension-4 form.
problem Characterize dimension reduction for moduli spaces of generalized ASD instantons.
method Integrability results for families of connections, topological criteria, and explicit descriptions of moduli spaces.
result Complete characterization of dimension reduction for moduli spaces of generalized ASD instantons.
There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three appr…
We review how a reduction procedure along a principal fibration and an unfolding procedure associated to a suitable momentum map allow to describe the Kähler geometry of a finite dimensional complex projective spaces.
The paper proves rigidity theorems for forms on reductive symmetric spaces.
problem Local rigidity of forms on reductive symmetric spaces under representations of discrete groups.
method General local rigidity theorem for pull-backs of homogeneous forms, reinterpretation of old results.
result Volume of closed manifolds is constant under deformation of -structure.
We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generali…
Reduces path integrals for interacting systems using dependent coordinates.
problem Reducing path integrals for systems with symmetry.
method Reduction procedure based on Wiener-type path integral, optimal nonlinear filtering, and projection of mean curvature vector field.
result Shows non-invariance of the measure in the path integral under reduction and generates the Jacobian.
Symplectic reduction by abelian subgroups coincides under specific conditions.
problem Conditions for symplectic reduction by abelian subgroups to match.
method Reduction-by-stages framework, nilpotent Lie groups, generic momentum values.
result Symplectomorphic reduced spaces under mild conditions.
The Landau-Lifshitz equation is derived as the reduction of a geodesic flow on the group of maps into the rotation group. Passing the symmetries of spatial isotropy to the reduced space is an example of semidirect product reduction by stages.
Reductive quotients preserve klt singularities in algebraic geometry.
problem Preserving klt singularities in quotients of klt singularities.
method Proving that the quotient of a klt type singularity by a reductive group is of klt type.
result The quotient of a klt variety by a reductive group results in a klt variety with a suitable boundary.
A new method reduces high-dimensional parameter spaces for faster numerical tasks.
problem Efficiently reducing high-dimensional parameter spaces for numerical tasks.
method Local Active Subspaces (LAS) combining active subspaces with clustering techniques.
result Significant speed-up in numerical tasks through efficient dimension reduction.
Sharpness of actions on reductive homogeneous spaces proven for various groups.
problem Proving proper and cocompact actions on reductive homogeneous spaces.
method Using quasi-isometric embedding and Anosov representations.
result Characterization and proof of non-compactness for certain homogeneous spaces.
This paper develops a Hamiltonian reduction method for field theories over affine principal bundles.
problem Developing a Hamiltonian reduction theory for field theories over affine principal bundles.
method Introducing a canonical identification to describe the reduced multisymplectic space without a connection.
result Derivation of reduced Hamilton-Cartan equations and a reduced covariant bracket.
In this paper, from the viewpoint of completeness of Marsden-Weinstein reduction, we illustrate how to give the definitions of a controlled Hamiltonian (CH) system and a reducible controlled Hamiltonian system with symmetry; and how to describe the dynamics of a CH system and the controlled Hamiltonian equivalence; as …
Geometrically revisits and models homogeneous spaces of compact Lie group .
problem Classifying homogeneous reductive spaces of compact Lie group .
method Geometrical approach to revisit and model the spaces.
result Explicit relations among geometric models of the spaces.
We extend the result in J. Reine Angew. Math. 664, 29-53, to the non-compact case. Namely, we prove that the canonical connection on a simply connected and irreducible naturally reductive space is unique, provided the space is not a sphere, a compact Lie group with a bi-invariant metric or its symmetric dual. In partic…
Survey of recent developments in symmetric reductions and controls for Hamiltonian systems.
problem Understanding the internal relationships of geometric structures and controls in Hamiltonian systems with symmetry.
method Survey and introduction of recent developments in controlled Hamiltonian systems with symmetry.
result Reveals the relationships between geometric structures, nonholonomic constraints, dynamical vector fields, and controls.