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.
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48 results for “Riemannian naturally reductive spaces”
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…
We provide examples of naturally reductive pseudo-Riemannian spaces, in particular an example of a naturally reductive pseudo-Riemannian 2-step nilpotent Lie group , such that is invariant under a left action and for which the center is degenerate. The metric does not correspond to a bi-in…
Tangent Lie groups have a special Riemannian metric structure.
problem Characterizing the Riemannian structure of tangent Lie groups.
method Proving the existence of a left-invariant naturally reductive metric and connection on tangent Lie groups.
result Tangent Lie groups admit a special Riemannian metric structure.
The paper explores Lorentzian connections with parallel skew torsion.
problem Understanding metric connections with parallel skew-symmetric torsion in Lorentzian signature.
method Analyzing holonomy algebras, torsion, and curvature; constructing examples; classifying homogeneous spaces.
result Complete classification of Lorentzian naturally reductive homogeneous spaces in low dimensions.
Geodesic orbit property studied for Lorentz manifolds.
problem Geodesic orbit property for Lorentz manifolds.
method Defined naturally reductive for pseudo-Riemannian manifolds and proved theorems for Lorentz nilmanifolds.
result Geodesic orbit property holds for Lorentz nilmanifolds under specific conditions.
Jacobi operators on certain naturally reductive spaces have constant coefficient ODEs.
problem Understanding geometric properties of naturally reductive spaces.
method Analyzing Jacobi operators and their ODEs on these spaces.
result Jacobi operators on these spaces satisfy constant coefficient ODEs.
Geodesic orbit spaces and their families are studied in pseudo-Riemannian manifolds.
problem Understanding geodesic orbit spaces and their properties in pseudo-Riemannian manifolds.
method Analyzing real form families of pseudo-Riemannian manifolds and proving properties of geodesic orbit spaces.
result Geodesic orbit spaces and their families have interesting properties in pseudo-Riemannian manifolds.
New definition of naturally reductive Finsler manifolds using geodesic graphs.
problem Defining naturally reductive Finsler manifolds using geodesic graphs.
method Proposed a new geometrical definition using geodesic graphs and constructed examples of Finsler metrics.
result Explicit examples of Finsler naturally reductive metrics constructed.
Naturally reductive metricsmath.DG
Study characterizes naturally reductive metrics on homogeneous manifolds.
problem Characterizing naturally reductive metrics on homogeneous manifolds.
method Characterization through local -products and equivalence of properties.
result Explicit flag curvature formula for naturally reductive metrics.
This paper deals with naturally reductive pseudo-Riemannian 2-step nilpotent Lie groups $(N, \la \,,\,\ra_N)$, such that $\la \,,\,\ra_N$ is invariant under a left action. The case of nondegenerate center is completely characterized. In fact, whenever $\la \,,\, \ra_N$ restricts to a metric in the center it is proved h…
Study invariant metrics on Ledger-Obata spaces, classifying and constructing them.
problem Classify and construct invariant metrics on Ledger-Obata spaces.
method Classification and explicit construction of naturally reductive metrics, proving geodesic orbit properties.
result A Ledger-Obata space is geodesic orbit if and only if the metric is naturally reductive.
The paper classifies geodesic orbit spaces with simple isotropy groups.
problem Classifying geodesic orbit spaces with simple isotropy groups.
method Classifying -naturally reductive and -geodesic orbit metrics on .
result Classification of geodesic orbit spaces with simple isotropy groups.
Classifies 7D manifolds with specific geometric properties.
problem Classifying 7D manifolds with parallel skew-symmetric torsion and holonomy.
method Extending Friedrich's work, using classification techniques for naturally reductive spaces and nearly parallel -structures.
result Complete classification of 7D manifolds with the specified properties.
Paper proves naturally reductive property is inaudible for certain manifolds.
problem Proving naturally reductive property is inaudible for specific manifolds.
method Using a pair of non-compact 11-dimensional generalized Heisenberg groups, the paper proves the naturally reductive property is inaudible.
result The naturally reductive property is inaudible for certain manifolds.
We present a new method for classifying naturally reductive homogeneous spaces -- i.\,e.~homogeneous Riemannian manifolds admitting a metric connection with skew torsion that has parallel torsion \emph{and} curvature. This method is based on a deeper understanding of the holonomy algebra of connections with parallel sk…
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.
Geodesic orbit nilmanifolds linked to graph structures.
problem Characterizing Riemannian nilmanifolds associated with graphs.
method Analyzing naturally reductive and geodesic orbit properties of nilmanifolds defined by graphs.
result Nilmanifolds are geodesic orbit if and only if naturally reductive if and only if the defining graph is a disjoint union of complete graphs.
Classifies geodesic orbit spaces with abelian isotropy subgroups.
problem Characterizing and classifying geodesic orbit spaces with specific isotropy subgroups.
method Simplified study of geodesic orbit metrics on G/S by reducing to submanifolds and generalized flag manifolds, using properties of root systems.
result Geodesic orbit spaces of the form (G/S,g) are naturally reductive.
Paper shows spectra can't distinguish naturally reductive manifolds.
problem Cannot distinguish naturally reductive manifolds using Laplace-Beltrami spectrum.
method Characterized naturally reductive 2-step nilpotent Lie groups via Ambrose-Singer's structures; constructed isospectral pairs of 9-dimensional nilmanifolds.
result Spectra of Laplace-Beltrami operator can't distinguish naturally reductive manifolds from non-naturally reductive ones.
Homogeneous three-spheres have only homogenous foliations.
problem Characterize foliations of homogeneous three-spheres.
method Prove that a three-sphere's metric foliations are homogenous if and only if it is naturally reductive.
result Homogeneous three-spheres have only homogenous foliations.
We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type…
Classifies 7- and 8-dimensional naturally reductive spaces.
problem Classifying naturally reductive spaces in 7 and 8 dimensions.
method Combines structure theory and new construction methods.
result Complete classification of 7- and 8-dimensional naturally reductive spaces.
New naturally reductive spaces constructed with group isometries.
problem Creating new naturally reductive spaces.
method Constructing infinitesimal models, specifying transitive groups of isometries, and explicitly defining the naturally reductive structure.
result A large number of new naturally reductive spaces constructed.
Flag manifolds are in general not symmetric spaces. But they are provided with a structure of -symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. We detail for the flag manifold what are the conditions…
The Pontryagin forms on 1-jet bundle of Riemannian metrics, are shown to provide, in a natural way, diffeomorphism-invariant pre-symplectic structures on the space of Riemannian metrics for dimensions . The equivariant Pontryagin forms provide canonical moment maps for these structures. In dimension two, the sy…
A characterization of maximal domains of existence of adapted complex structures for Riemannian homogeneous manifolds under certain extensibility assumptions on their geodesic flow is given. This is applied to generalized Heisenberg groups and naturally reductive Riemannian homogeneous spaces. As an application it is s…
In this paper we consider invariant Matsumoto metrics which are induced by invariant Riemannian metrics and invariant vector fields on homogeneous spaces then we give the flag curvature formula of them. Also we study the special cases of naturally reductive spaces and bi-invariant metrics. We end the article by giving …
Totally geodesic submanifolds of the homogeneous nearly Kähler 6-manifolds and their G2-conesmath.DG
Classifies totally geodesic submanifolds in specific geometric spaces.
problem Identifying totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
method Developed new techniques for studying totally geodesic submanifolds in analytic Riemannian manifolds, homogeneous spaces, and Riemannian cones.
result Obtained a classification of totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
Investigates solving curvature equations on special Lie groups.
problem Solving curvature equations on non-compact simple Lie groups.
method Analyzes left-invariant naturally reductive metrics and conditions for solvability.
result Obtains conditions for the solvability of curvature equations.
Paper constructs naturally reductive spaces with a general formula.
problem Understanding naturally reductive spaces.
method Explicit construction from \cite{Storm2018} and general formula derivation.
result Proves reducibility and isomorphism criteria.
The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by J. E. D'Atri and W. Ziller in 1979. In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for , by using a method of Riemannian submersions. In the pres…
Study of symmetry distributions in Lorentzian naturally reductive nilmanifolds.
problem Understanding symmetry in Lorentzian naturally reductive nilmanifolds.
method Analysis of 2-step nilpotent Lorentzian Lie groups with transitive isometry subgroups.
result Fixed points of isotropy representation indicate the distribution of symmetry.
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 …
This work concerns the non-flat metrics on the Heisenberg Lie group of dimension three $\Heis_3(\RR)$ and the bi-invariant metrics on the solvable Lie groups of dimension four. On $\Heis_3(\RR)$ we prove that the property of the metric being naturally reductive is equivalent to the property of the center being non-dege…
A novel Riemannian extension of stochastic variance reduction for manifold optimization.
problem Optimization on the Grassmann manifold for large-scale problems.
method Riemannian stochastic variance reduced gradient (R-SVRG) on the Grassmann manifold.
result The proposed algorithm outperforms standard Riemannian SGD on various problems.
A very important class of homogeneous Riemannian manifolds are the so-called normal homogeneous spaces, which have associated a canonical connection. In this work we obtain geometrically the (connected component of the) group of affine transformations with respect to the canonical connection for a normal homogeneous sp…
Proposes ridge regression on Riemannian manifolds for time-series prediction.
problem Time-series prediction on Riemannian manifolds.
method Combines Riemannian least-squares fitting via Bézier curves, empirical covariance on manifolds, and Mahalanobis distance regularization.
result Significant error reduction in synthetic spherical experiments and hurricane forecasting.
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 geodesic orbit Lorentz nilmanifolds, proving structural properties.
problem Understanding the structure of geodesic orbit Lorentz nilmanifolds.
method Analyzing geodesic orbit Lorentz nilmanifolds with reductive decompositions.
result Proves properties of nilpotent subgroups and their nilpotency steps.
Given a reductive homogeneous space M=G/H endowed with a naturally reductive metric, we study the one-parameter family of connections joining the canonical and the Levi-Civita connection (t=0, 1/2). We show that the Dirac operator D^t corresponding to t=1/3 is the so-called ``cubic'' Dirac operator recently introduced …
A new Riemannian algorithm reduces variance in manifold optimization.
problem Optimizing functions on manifolds with stochastic gradient descent.
method Riemannian stochastic variance reduction with retraction and vector transport.
result The proposed algorithm outperforms standard methods on SPD and Grassmann manifolds.
The paper studies geometries with parallel skew-symmetric torsion and their submersions.
problem Understanding geometries with parallel skew-symmetric torsion.
method Analyzing metric connections and submersions.
result Complete local classification of geometries with parallel skew-symmetric torsion in principal bundle cases.
Any Spin(7)-manifold admits a metric connection \nabla^c with totally skew-symmetric torsion T^c preserving the underlying structure. We classify those with \nabla^c-parallel T^c\neq0 and non-Abelian isotropy algebra iso(T^c)<spin(7). These are isometric to either Riemannian products or homogeneous naturally reductive …
Study new symmetries in non-symmetric spaces and discontinuous groups.
problem Analyze symmetries in non-symmetric homogeneous spaces and discontinuous groups.
method Investigate discrete series, discontinuous groups, and analysis on pseudo-Riemannian spaces.
result New insights into symmetries of non-symmetric homogeneous spaces and discontinuous groups.
The paper studies symmetry reduction and optimal control on Riemannian manifolds.
problem Symmetry reduction and optimal control on Riemannian manifolds.
method Derivation of reduced equations of motion for variational problems on Lie groups and Riemannian homogeneous spaces.
result Derivation of geodesic equations and reduced equations of motion for various applications.
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.
Study on special Lie groups with Lorentzian metrics.
problem Characterize structure of -step nilpotent Lorentzian naturally reductive Lie groups.
method Develop framework for naturally reductive Lie groups, extend to Lorentzian context, analyze degenerate and non-degenerate cases.
result Complete structural description of naturally reductive -step Lorentzian nilpotent Lie groups.