Complete classification of quaternionic skew-Hermitian symmetric spaces found.
problem Classifying quaternionic skew-Hermitian symmetric spaces.
method Proving the existence of a torsion-free mSO∗(2n)mSp(1)-structure and showing that any homogeneous space is symmetric. result A complete classification of quaternionic skew-Hermitian symmetric spaces for arbitrary n>1. Study various submanifolds in quaternionic skew-Hermitian spaces.
problem Characterize submanifolds in almost quaternionic skew-Hermitian manifolds.
method Construct explicit examples of submanifolds in semisimple quaternionic skew-Hermitian symmetric spaces.
result Explicit examples of submanifolds for each type considered.
This paper studies geometric structures on manifolds with specific symplectic properties.
problem Understanding geometric structures on manifolds with quaternionic skew-Hermitian properties.
method Equivalent definitions, intrinsic torsion, classification of geometries, explicit connections.
result Classification of symmetric spaces with invariant torsion-free structures.
The paper characterizes curvature of quaternionic skew-Hermitian manifolds and constructs related geometric structures.
problem Characterizing the curvature of quaternionic skew-Hermitian manifolds.
method Holonomy theory of symplectic connections and bundle constructions.
result Existence and integrability of almost hypercomplex skew-Hermitian structures on Swann bundles.
In this paper we generalize the known DDVV-type inequalities for real (skew-)symmetric and complex (skew-)Hermitian matrices to arbitrary real, complex and quaternionic matrices. Inspired by the Erdős-Mordell inequality, we establish the DDVV-type inequalities for matrices in the subspaces spanned by a Clifford system …
In this paper we extend DDVV-type inequalities involving the Frobenius norm of commutators from real symmetric and skew-symmetric matrices to Hermitian and skew-Hermitian matrices.
The study finds conditions for quaternionic structures on symmetric spaces.
problem Conditions for quaternionic structures on symmetric spaces.
method Analysis of Lie group actions and representations.
result Symmetric spaces have invariant quaternionic structures under specific conditions.
New theorem links quaternionic-Kähler manifolds to symmetric spaces.
problem Understanding curvature properties of quaternionic-Kähler manifolds.
method Analyzing positive scalar curvature and nonnegative sectional curvature.
result Compact quaternionic-Kähler manifolds with these properties are symmetric.
Paper constructs an example of a non-compact submanifold in a quaternionic Kähler symmetric space.
problem Tackles the construction of a non-compact totally complex submanifold in a quaternionic Kähler symmetric space.
method Uses an isometric action of a compact Lie group and a maximal totally geodesic sphere.
result Proves the existence of a non-compact totally complex submanifold of maximal dimension in a compact quaternionic Kähler symmetric space.
The paper extends Busemann's inequalities to complex and quaternionic spaces.
problem Extending Busemann's inequalities to complex and quaternionic vector spaces.
method Proof leverages a monotonicity property under symmetrization with respect to complex or quaternionic hyperplanes.
result Standard Steiner symmetrization does not exhibit the monotonicity property in complex or quaternionic spaces.
This is a complete classification of the complex forms of quaternionic symmetric spaces
New examples of hypersurfaces found in quaternionic hyperbolic spaces.
problem Classifying actions on symmetric spaces of rank one.
method Cohomogeneity one actions and orbit equivalence.
result Uncountably many inhomogeneous isoparametric families of hypersurfaces.
Study finds only two CROSSes can have specific quaternionic structure.
problem Characterizing compact rank one symmetric spaces with quaternionic structures.
method Analyzing properties of CROSSes (compact rank one symmetric spaces).
result Only HPn and CP2 admit almost quaternionic structures. We show that pseudo-Riemannian almost quaternionic homogeneous spaces with index 4 and an H-irreducible isotropy group are locally isometric to a pseudo-Riemannian quaternionic Kähler symmetric space if the dimension is at least 16. In dimension 12 we give a non-symmetric example.
The paper extends Gray's result to quaternion-Kähler manifolds.
problem Understanding quaternion-Kähler manifolds with non-negative quaternionic sectional curvature.
method Introducing quaternionic sectional curvature, proving Wolf spaces have non-negative curvature, and using nearly Kähler twistor spaces.
result Every quaternion-Kähler manifold with non-negative quaternionic sectional curvature is a Wolf space.
We analyze polar actions on Hermitian and quaternion-Kähler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classificatio…
Constructs explicit harmonic functions and morphisms on complex and quaternionic Grassmannians.
problem Creating explicit solutions for p-harmonic functions and harmonic morphisms. method Using joint eigenfunctions of the Laplace-Beltrami and conformality operators.
result Induces solutions on dual non-compact Riemannian symmetric spaces.
We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian manifolds. We classify all structures with such symmetry dimensions. Geometric propertie…
We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to the quaternionic p…
We prove that compact quaternionic-Kähler manifolds of positive scalar curvature admit no almost complex structure, even in the weak sense, except for the complex Grassmannians Gr2(Cn+2). We also prove that irreducible inner symmetric spaces M4n of compact type are not weakly complex, except for spheres and …
Unified framework for complex-valued eigenfunctions on Riemannian symmetric spaces.
problem Finding a unified scheme for complex-valued eigenfunctions on Riemannian symmetric spaces.
method Employing the Cartan embedding for classical compact Riemannian symmetric spaces and quaternionic Grassmannians.
result Construction of new eigenfunctions on quaternionic Grassmannians.
Study of geometric structures on manifolds, focusing on integrability conditions.
problem Understanding the integrability of specific geometric structures.
method Analysis of algebraic types, intrinsic torsions, and distinguished connections.
result Presented first-order integrability conditions and geometric interpretations.
We classify non-polar irreducible representations of connected compact Lie groups whose orbit space is isometric to that of a representation of a finite extension of Sp(1)k for some k>0. It follows that they are obtained from isotropy representations of certain quaternion-Kähler symmetric spaces by restricting to …
Positive Quaternion Kaehler Manifolds are Riemannian manifolds with holonomy contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they are symmetric spaces. We prove this conjecture in dimension 20 under additional assumptions and we provide recognition theorems for quaternionic projective spaces …
Study on quaternionic bisectional curvature for quaternion-Kähler manifolds.
problem Characterize quaternionic bisectional curvature on quaternion-Kähler manifolds.
method Analyzing properties of quaternionic bisectional curvature on specific manifolds.
result Non-negative quaternionic bisectional curvature is only on quaternionic projective space.
The study classifies hypersurfaces in quaternionic space forms with constant principal curvatures.
problem Classifying hypersurfaces in quaternionic space forms with specific curvature properties.
method Analyzing curvature-adapted real hypersurfaces in non-flat quaternionic space forms HPm and HHm. result Classification of hypersurfaces including geodesic hyperspheres, tubes, and specific examples in HPm and HHm. Study on stability of Einstein metrics on symmetric spaces.
problem Stability of Einstein-Hilbert functional on compact symmetric spaces.
method Classification of irreducible representations and use of Casimir eigenvalues.
result Proves stability of Einstein metrics on quaternionic and Cayley projective plane, instability on other quaternionic Grassmannians.
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}.
Possible holonomy algebras of pseudo-quaternionic-Kählerian manifolds of signature (4,4) are classified. Using this, a new proof of the classification of simply connected pseudo-quaternionic-Kählerian symmetric spaces of signature (4,4) is obtained.
We classify indefinite simply connected hyper-Kaehler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m,4m). We establish a natural 1-1 correspondence between simply connected hyper-Kaehler symmetric spaces of dimension 8m and orbits of the general linear group GL(m,H…
The study classifies homogeneous Sasaki manifolds over quaternionic Kähler spaces.
problem Classifying homogeneous Sasaki manifolds over quaternionic Kähler spaces.
method Locally defined Riemannian submersions and homogeneous space constructions.
result Complete classification of homogeneous Sasaki manifolds in the non-degenerate case.
In a previous article we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kaehler manifolds. In the present article we study the limiting case, i. e. manifolds where the lower bound is attained as an eigenvalue. We give an equivalent formulation in terms of a quaternionic Killing equation and…
We develop and study quaternionic and octonionic analogies of Cartan angular and Toledo invariants that are well known in the complex hyperbolic space. Using such invariants we study quasifuchsian deformations (including bendings) of quaternionic and octonionic hyperbolic manifolds.
Paper proves injectivity of non-abelian X-ray transform on certain spaces.
problem Injectivity of non-abelian X-ray transform on asymptotically hyperbolic spaces.
method Gauge equivalence for unitary connections and skew-Hermitian Higgs fields.
result Injectivity result for non-abelian X-ray transform over skew-Hermitian Higgs fields.
Maps Lagrangian submanifolds to quaternionic projective space, proving one-to-one correspondences.
problem Constructing maps between Lagrangian submanifolds and quaternionic projective spaces.
method Explicit construction of maps from minimal δ(2)-ideal Lagrangian submanifolds of Cn to HPn−1. result One-to-one correspondences between minimal Lagrangian surfaces in CP2 and minimal totally complex surfaces in HP2. We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces of dimension greater than two. For the quaternionic hyperbolic spaces of dimension greater than two we reduce the classification problem to…
Researchers found non-Killing tensor fields on certain symmetric spaces.
problem Understanding Killing tensors on all Riemannian symmetric spaces.
method Constructed explicit examples of quadratic Killing tensors on quaternionic and Cayley projective spaces.
result Quadratic Killing tensors can be non-Killing on some symmetric spaces.
The Riemannian symmetric space SU_{2,m}/S(U_2U_m) is both Hermitian symmetric and quaternionic Kahler symmetric. Let M be a hypersurface in SU_{2,m}/S(U_2U_m) and denote by TM its tangent bundle. The complex structure of SU_{2,m}/S(U_2U_m) determines a maximal complex subbundle C of TM, and the quaternionic structure o…
Finite volume ends found in quaternionic Kähler manifolds.
problem Existence of quaternionic Kähler manifolds with finite volume ends.
method Proof in all dimensions 4m≥4. result Existence of quaternionic Kähler manifolds with finite volume ends.
Harmonic morphisms and p-harmonic functions constructed on symmetric spaces.
problem Constructing harmonic morphisms and p-harmonic functions on symmetric spaces.
method Using Cartan embedding and related maps to relate tension field and conformality operator.
result Simple formulae relating tension field and conformality operator on symmetric spaces to those on their images.
The paper derives inequalities for submanifolds in quaternionic Kaehler manifolds.
problem Analyzing submanifolds in quaternionic Kaehler manifolds.
method Established Chen's and generalized Casorati curvature inequalities.
result Derived inequalities for submanifolds in quaternionic Kaehler manifolds.
Researchers prove spectral uniqueness of complex/quaternionic structures on manifolds.
problem Spectral uniqueness of complex/quaternionic structures on manifolds.
method Explicit expression for smallest positive eigenvalue of Laplace-Beltrami operator.
result Irreducible symmetric spaces are spectrally unique within families of homogeneous metrics.
This paper establishes inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.
problem Establishing higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.
method Developing factorization theorems and introducing Geller's operators, combining with Helgason-Fourier analysis and kernel estimates.
result Established higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.
Let V be the pseudo-Euclidean vector space of signature (p,q), p>2 and W a module over the even Clifford algebra Cl^0 (V). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V)-equivariant linear map Π: \wedge^2 W \to V. If the skew symmetric vector valued bilinear form Πis nondegenerate then (M,Q) i…
The symmetry dimension of a geometric structure is the dimension of its symmetry algebra. We investigate symmetries of almost quaternionic structures of quaternionic dimension n. The maximal possible symmetry is realized by the quaternionic projective space HPn, which is flat and has the symmetry algebra …
Researchers create non-homogeneous finite-volume ends on quaternionic Kähler manifolds.
problem Constructing non-homogeneous quaternionic Kähler manifolds with finite volume ends.
method Using cohomogeneity one deformation of symmetric spaces, the researchers constructed manifolds with specific fundamental groups.
result The constructed manifolds are aspherical and have finite volume ends, not locally homogeneous.
The paper studies 8D manifolds with a specific tensor field called a cubic discriminant.
problem Characterizing and understanding 8D Riemannian manifolds with reduced structure groups.
method Introducing an almost quaternion-Hermitian structure and a cubic discriminant tensor field.
result Only two non-flat, integrable examples of these structures are found: quaternion-Kähler symmetric spaces.
Method calculates spectra of Rarita-Schwinger operator on symmetric spaces.
problem Calculating spectra of the Rarita-Schwinger operator on compact symmetric spaces.
method Using Weitzenböck formulas, Laplace operator, Casimir operator, Freudenthal's formula, and branching rules.
result Obtained spectra on the sphere, complex projective space, and quaternionic projective space.