Quaternionic Manifolds Research Articles

Cohomology rings of the real and oriented partial flag manifolds

We show, in contrast to the cases of complex and quaternionic partial flag manifolds, the rational cohomology rings of the real and oriented partial flag manifolds are generated not only by the characteristic classes of their canonical vector bundles, but also by exterior algebras arising from the rational cohomology rings of real Stiefel manifolds. We also discuss the rational and the mod-2 equivariant cohomology rings of these partial flag manifolds for certain canonical actions of tori and 2-tori.

Improved Pose Graph Optimization for Planar Motions Using Riemannian Geometry on the Manifold of Dual Quaternions

Abstract We present a novel Riemannian approach for planar pose graph optimization problems. By formulating the cost function based on the Riemannian metric on the manifold of dual quaternions representing planar motions, the nonlinear structure of the SE(2) group is inherently considered. To solve the on-manifold least squares problem, a Riemannian Gauss– Newton method using the exponential retraction is applied. The proposed Riemannian pose graph optimizer (RPG-Opt) is further evaluated based on public planar pose graph data sets. Compared with state-of-the-art frameworks, the proposed method gives equivalent accuracy and better convergence robustness under large uncertainties of odometry measurements.

Hyperspherical Unscented Particle Filter for Nonlinear Orientation Estimation

We propose a novel quaternion particle filter for nonlinear SO(3) estimation. For importance sampling, the proposal distribution is designed to incorporate newly observed evidence. For that, the unscented Kalman filtering is performed particle-wise on the tangent plane of the unit quaternion manifold via gnomonic projection/retraction based on hyperspherical geometry. As prior particles are driven towards high-likelihood regions on the manifold, computational efficiency of quaternion particle filtering is significantly improved. The resulting hyperspherical unscented particle filter (HUPF) is evaluated for nonlinear orientation estimation in simulations. Results show that it gives superior tracking performance compared with the conventional particle filter and other existing quaternion filtering schemes relying on parametric modeling.

On Compact Affine Quaternionic Curves and Surfaces

This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira Theorem that studies compact complex manifolds in complex dimension 2, states that the only compact affine quaternionic curves are the quaternionic tori and the primary Hopf surface $$S^3\times S^1$$ . As for compact affine quaternionic surfaces, we restrict to the complete ones: the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex simply connected 8-dimensional Lie Groups, identifies a path towards their classification.

Relative singular value decomposition and applications to LS-category

Let Sp(n) be the symplectic group of quaternionic (n×n)-matrices. For any 1≤k≤n, an element A of Sp(n) can be decomposed in A=[αTβP] with P a (k×k)-matrix. In this work, starting from a singular value decomposition of P, we obtain what we call a relative singular value decomposition of A. This feature is well adapted for the study of the quaternionic Stiefel manifold Xn,k, and we apply it to the determination of the Lusternik-Schnirelmann category of Sp(k) in X2k−j,k, for j=0,1,2.

Examples of transversally complex submanifolds of the associative Grassmann manifold

We consider the Grassmann manifold of associative subspaces in the space of imaginary octonions, which we call the associative Grassmann manifold. It is known that the associative Grassmann manifold is an eight-dimensional compact symmetric quaternionic Kahler manifold. We construct interesting examples of four-dimensional complex submanifolds of the associative Grassmann manifold.

On C $$\varvec{\otimes }$$ H $$\varvec{\otimes }$$ O-Valued Gravity, Sedenions, Hermitian Matrix Geometry and Nonsymmetric Kaluza–Klein Theory

We review briefly how R $$\otimes $$ C $$\otimes $$ H $$\otimes $$ O-valued gravity (real-complex-quaterno-octonionic gravity) naturally can describe a grand unified field theory of Einstein’s gravity with a Yang–Mills theory containing the Standard Model group $$SU(3) \times SU(2) \times U(1)$$ . The algebra of left actions associated with the composite algebras involving the Division algebras, and the Sedenions $$\mathbf{S}$$ , and acting on themselves, all lead to complex Clifford algebras (complex matrix algebras). The complex Cl(16) algebra is the most appealing one since it is the one corresponding to the algebra of left actions of $$\mathbf{C} \otimes \mathbf{H} \otimes \mathbf{O} \otimes \mathbf{S}$$ acting on itself, and containing the $$\mathbf{e_8 \oplus e_8}$$ algebra of the anomaly free 10D Heterotic String. An analysis of $$ C \otimes H \otimes O$$ -valued gravity reveals that it bears a connection to Nonsymmetric Kaluza–Klein theories and complex Hermitian Matrix Geometry. The key behind these connections is in finding the relation between $$ C \otimes H \otimes O$$ -valued metrics in two complex dimensions with complex metrics in higher dimensional real manifolds ( $$ D = 32 $$ real dimensions in particular). It is desirable to extend these results to hypercomplex, quaternionic manifolds and Exceptional Jordan Matrix Models.

Harmonic morphisms and the Penrose–Ward transform

We show that, in quaternionic geometry, the Ward transform is a manifestation of the functoriality of the basic correspondence between the \rho -quaternionic manifolds and their twistor spaces. We apply this fact, together with the Penrose transform, to obtain existence results for quaternionic manifolds and for harmonic morphisms from hyper-Kähler manifolds.

A Bonnet–Myers type theorem for quaternionic contact structures

We prove a Bonnet–Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifold is compact and we give a sharp bound on its sub-Riemannian diameter.

Projective geometry and the quaternionic Feix–Kaledin construction

Starting from a complex manifold S with a real-analytic c-projective structure whose curvature has type (1,1), and a complex line bundle L with a connection whose curvature has type (1,1), we construct the twistor space Z of a quaternionic manifold M with a quaternionic circle action which contains S as a totally complex submanifold fixed by the action. This extends a construction of hypercomplex manifolds, including hyperkaehler metrics on cotangent bundles, obtained independently by B. Feix and D. Kaledin. When S is a Riemann surface, M is a self-dual conformal 4-manifold, and the quotient of M by the circle action is an Einstein-Weyl manifold with an asymptotically hyperbolic end, and our construction coincides with a construction presented by the first author in a previous paper. The extension also applies to quaternionic Kaehler manifolds with circle actions, as studied by A. Haydys and N. Hitchin.

Quaternionic toric manifolds

In the present paper we introduce and study a new notion of toric manifold in the quaternionic setting. We develop a construction with which, starting from appropriate $m$-dimensional Delzant polytopes, we obtain manifolds of real dimension $4m$, acted on by $m$ copies of the group ${\rm Sp}(1)$ of unit quaternions. These manifolds are quaternionic regular and can be endowed with a $4$-plectic structure and a generalized moment map. Convexity properties of the image of the moment map are studied. Quaternionic toric manifolds appear to be a large enough class of examples where one can test and study new results in quaternionic geometry.

New homogeneous Einstein metrics on quaternionic Stiefel manifolds

Abstract We consider invariant Einstein metrics on the quaternionic Stiefel manifold V p ℍ n of all orthonormal p-frames in ℍ n . This manifold is diffeomorphic to the homogeneous space Sp(n)/Sp(n − p) and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on V p ℍ n ≅ Sp(n)/Sp(n − p), where n = k 1 + k 2 + k 3 and p = n − k 3. We view V p ℍ n as a total space over the generalized Wallach space Sp(n)/(Sp(k 1)×Sp(k 2)×Sp(k 3)) and over the generalized flag manifold Sp(n)/(U(p)×Sp(n − p)).

The qc Yamabe problem on non-spherical quaternionic contact manifolds

It is shown that the qc Yamabe problem has a solution on any compact qc manifold which is non-locally qc equivalent to the standard 3-Sasakian sphere. Namely, it is proved that on a compact non-locally spherical qc manifold there exists a qc conformal qc structure with constant qc scalar curvature.

On quaternionic tori and their moduli space

Quaternionic tori are defined as quotients of the skew field \mathbb H of quaternions by rank-4 lattices. Using slice regular functions, these tori are endowed with natural structures of quaternionic manifolds (in fact quaternionic curves), and a fundamental region in a 12-dimensional real subspace is then constructed to classify them up to biregular diffeomorphisms. The points of the moduli space correspond to suitable special bases of rank-4 lattices, which are studied with respect to the action of the group GL (4, \mathbb Z) , and up to biregular diffeomeorphisms. All tori with a non trivial group of biregular automorphisms – and all possible groups of their biregular automorphisms – are then identified, and recognized to correspond to five different subsets of boundary points of the moduli space.

Noncommutative products of Euclidean spaces

We present natural families of coordinate algebras of noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the Yang-Baxter equations. As a consequence they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces which are particularly well behaved and are deformations parametrised by a two-dimensional sphere. Quotients include noncommutative seven-spheres as well as noncommutative "quaternionic tori". There is invariance for an action of $SU(2) \times SU(2)$ in parallel with the action of $U(1) \times U(1)$ on a "complex" noncommutative torus which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.

PROJECTIVE STRUCTURES AND -CONNECTIONS

We extend T. Y. Thomas’s approach to projective structures, over the complex analytic category, by involving the$\unicode[STIX]{x1D70C}$-connections. This way, a better control of projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold$P$is endowed with a complex projective structure then$P$can be locally identified, through quaternionic diffeomorphisms, with the quaternionic projective space.

On quaternionic complexes over unimodular quaternionic manifolds

Penrose's two-spinor notation for 4-dimensional Lorentzian manifolds is extended to two-component notation for quaternionic manifolds, which is a useful tool for calculation. We can construct a family of quaternionic complexes over unimodular quaternionic manifolds only by elementary calculation. On complex quaternionic manifolds as complexification of quaternionic Kähler manifolds, the existence of these complexes was established by Baston by using twistor transformations and spectral sequences. Unimodular quaternionic manifolds constitute a large nice class of quaternionic manifolds: there exists a very special curvature decomposition; the conformal change of a unimodular quaternionic structure is still unimodular quaternionic; the complexes over such manifolds are conformally invariant. This class of manifolds is the real version of torsion-free QCFs introduced by Bailey and Eastwood. These complexes are elliptic. We also obtain a Weitzenböck formula to establish vanishing of the cohomology groups of these complexes for quaternionic Kähler manifolds with negative scalar curvatures.

Almost h-conformal slant submersions from almost quaternionic Hermitian manifolds

We introduce the notions of h-conformal slant submersions and almost h-conformal slant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions, h-slant submersions, almost h-slant submersion and conformal slant submersions. We investigate several properties of these including the integrability of distributions, the geometry of foliations and the conditions for such maps to be totally geodesic. Further we give some examples of such maps.

Maximal radius of quaternionic hyperbolic manifolds

We derive an explicit lower bound on the radius of a ball embedded in a quaternionic hyperbolic manifold.

On Quaternionic 3 CR-Structure and Pseudo-Riemannian Metric

A CR-structure on a 2n +1-manifold gives a conformal class of Lorentz metrics on the Fefferman S1-bundle. This analogy is carried out to the quarternionic conformal 3-CR structure (a generalization of quaternionic CR- structure) on a 4n + 3 -manifold M. This structure produces a conformal class [g] of a pseudo-Riemannian metric g of type (4n + 3,3) on M × S3. Let (PSp(n +1,1), S4n+3) be the geometric model obtained from the projective boundary of the complete simply connected quaternionic hyperbolic manifold. We shall prove that M is locally modeled on (PSp(n +1,1), S4n+3) if and only if (M × S3 ,[g]) is conformally flat (i.e. the Weyl conformal curvature tensor vanishes).