Quaternionic Manifolds Research Articles

On h-Quasi-Hemi-Slant Riemannian Maps

In the present article, we indroduce and study h-quasi-hemi-slant (in short, h-qhs) Riemannian maps and almost h-qhs Riemannian maps from almost quaternionic Hermitian manifolds to Riemannian manifolds. We investigate some fundamental results mainly on h-qhs Riemannian maps: the integrability of distributions, geometry of foliations, the condition for such maps to be totally geodesic, etc. At the end of this article, we give two non-trivial examples of this notion.

Slice conformality and Riemann manifolds on quaternions and octonions

In this paper we establish quaternionic and octonionic analogs of the classical Riemann surfaces. The construction of these manifolds has nice peculiarities and the scrutiny of Bernhard Riemann approach to Riemann surfaces, mainly based on conformality, leads to the definition of slice conformal or slice isothermal parameterization of quaternionic or octonionic Riemann manifolds. These new classes of manifolds include slice regular quaternionic and octonionic curves, graphs of slice regular functions, the 4 and 8 dimensional spheres, the helicoidal and catenoidal 4 and 8 dimensional manifolds. Using appropriate Riemann manifolds, we also give a unified definition of the quaternionic and octonionic logarithm and n-th root function.

Halving spaces and lower bounds in real enumerative geometry

We develop the theory of halving spaces to obtain lower bounds in real enumerative geometry. Halving spaces are topological spaces with an action of a Lie group $\Gamma$ with additional cohomological properties. For $\Gamma=\mathbb{Z}_2$ we recover the conjugation spaces of Hausmann, Holm and Puppe. For $\Gamma=\mathrm{U}(1)$ we obtain the circle spaces. We show that real even and quaternionic partial flag manifolds are circle spaces leading to non-trivial lower bounds for even real and quaternionic Schubert problems. To prove that a given space is a halving space, we generalize results of Borel and Haefliger on the cohomology classes of real subvarieties and their complexifications. The novelty is that we are able to obtain results in rational cohomology instead of modulo 2. The equivariant extension of the theory of circle spaces leads to generalizations of the results of Borel and Haefliger on Thom polynomials.

On the systole growth in congruence quaternionic hyperbolic manifolds

We provide an explicit lower bound for the systole in principal congruence covers of compact quaternionic hyperbolic manifolds. We also prove the optimality of this lower bound.

INTEGRABILITY OF QUATERNION-KÄHLER SYMMETRIC SPACES

We find a necessary condition for the existence of an action of a Lie group G by quaternionic automorphisms on an integrable quaternionic manifold in terms of representations of 𝔤. We check this condition and prove that a Riemannian symmetric space of dimension 4n for n ≥ 2 has an invariant integrable almost quaternionic structure if and only if it is quaternionic vector space, quaternionic hyperbolic space, or quaternionic projective space.

On the cohomology rings of partially projective quaternionic Stiefel manifolds

Abstract The quaternionic Stiefel manifold is the total space of a fibre bundle over the corresponding Grassmannian . The group acts freely on the fibres of this bundle. The quotient space is called the quaternionic projective Stiefel manifold. Its real and complex analogues were actively studied earlier by a number of authors. A finite group acting freely on the three-dimensional sphere also acts freely and discretely on the fibres of the quaternionic Stiefel bundle. The corresponding quotient spaces are called partially projective Stiefel manifolds. The cohomology rings of partially projective quaternionic Stiefel manifolds with coefficients in , where is prime, are calculated. Bibliography: 14 titles.

On the almost h-conformal semi-invariant Riemannian maps

In this paper, we introduce hsR maps and ahsR maps from almost quaternionic Hermitian manifolds into Riemannian manifolds as a generalization of almost h-conformal semi-invariant submersions and conformal semi-invariant Riemannian maps. We give some examples of these maps and study their properties: the geometry of foliations, the integrability of distributions, the conditions for these maps to be harmonic or totally geodesic.

On the geometry of generalized almost quaternionic manifolds of vertical type

On the geometry of generalized almost quaternionic manifolds of vertical type

Efficient Trajectory Optimization for Constrained Spacecraft Attitude Maneuvers

This paper introduces an optimal trajectory planner for spacecraft rest-to-rest attitude transfers subject to input saturation, angular velocity, and nonconvex exclusion cone constraints. The proposed solution adapts the Projection-Operator-Based Newton’s Method for Trajectory Optimization (PRONTO) to address the nonlinearity of the unit quaternion manifold. The system constraints are then handled using a modified interior point method designed to remain applicable when an intermediate solution estimate is infeasible. Through an extensive numerical study, the proposed solution is compared with a state-of-the-art commercial solver, its computational efficiency is demonstrated, and its unique ability to provide feasible intermediate solutions is highlighted. These results strongly indicate PRONTO as a suitable real-time optimal maneuver planner for spacecraft attitude control.

On tame ρ-quaternionic manifolds

We introduce the notion of tame ρ-quaternionic manifold that permits the construction of a finite family of ρ-connections, significant for the geometry involved. This provides, for example, the following:• A new simple global characterisation of flat (complex-)quaternionic manifolds.• A new simple construction of the metric and the corresponding Levi-Civita connection of a quaternionic-Kähler manifold by starting from its twistor space; moreover, our method provides a natural generalization of this correspondence.Also, a new construction of quaternionic manifolds is obtained, and the properties of twistorial harmonic morphisms with one-dimensional fibres from quaternionic-Kähler manifolds are studied.

A Discrete Quaternion Particle Filter Based on Deterministic Sampling for IMU Attitude Estimation

A discrete quaternion filtering algorithm following high-density uncertainty matching-based deterministic sampling scheme is proposed for IMU attitude estimation. Existing quaternion filters rely on the specific distributions directly or the on-tangent-plane Gaussian distribution indirectly to model the underlying uncertainty. The key idea of this paper is to apply a Dirac mixture at few deterministic weighted samples for non-parametric interpreting the high-density uncertainty of unit quaternion manifold. In contrast to standard Monte Carlo techniques based on random sampling, the proposed approach is deterministic. Furthermore, the deterministic quaternion particles are obtained by projecting on-sphere uniform points to the unit quaternion manifold via an exponential map. The on-manifold quaternions are then exploited in a discrete Bayesian filtering procedure. We compare the new filter with the genetic algorithm embedded quaternion particle filter (GA-QPF) and the unscented quaternion estimator (USQUE) for accuracy and computation cost by using simulated and practical IMU data. The results illustrate that the proposed estimator possesses lower computational cost than the GA-QPF while providing a comparable accuracy, higher accuracy than the USQUE, as well as more effective against large initialization error.

A quaternion-based indirect Gaussian particle filter for nonlinear attitude estimation.

A recursively indirect quaternion estimator based on the Gaussian particle filter (GPF) is proposed for nonlinear attitude estimation. The key idea is to estimate an on-tangent-plane Gaussian distribution in the GPF scheme for interpreting the uncertainty of the unit quaternion manifold. The unit quaternion is provided with a global nonsingular attitude description in the prediction step, and the three-dimensional attitude error is estimated in the update step. Based on the framework of the GPF, the proposed filter does not need resampling and regularization compared with the PF. The performance of the proposed filter is verified theoretically and evaluated by experiments. The results show that the proposed filter has a faster convergence speed, lower complexity, and lower computational cost than the existing quaternion PF under a comparable accuracy.

Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces (M=G∕H,g) whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (Sp(n)∕Sp(n1)×⋯×Sp(ns),g), with 0<n1+⋯+ns≤n. Such spaces include spheres, quaternionic Stiefel manifolds, Grassmann manifolds and quaternionic flag manifolds. The present work is a contribution to the study of g.o. spaces (G∕H,g) with H semisimple.

Twistor space of a generalized quaternionic manifold

We first make a little survey of the twistor theory for hypercomplex, generalized hypercomplex, quaternionic or generalized quaternionic manifolds. This last theory was initiated by Pantilie (Ann. Mat. Pura. Appl. 193 (2014) 633–641), and allows one to extend the Penrose correspondence from the quaternion to the generalized quaternion case. He showed that any generalized almost quaternionic manifold equipped with an appropriate connection admit a twistor space which comes naturally equipped with a tautological almost generalized complex structure. But he has left open the problem of the integrability. The aim of this article is to give an integrability criterion for this generalized almost complex structure and to give some examples especially in the case of generalized hyperkähler manifolds using the generalized Bismut connection, introduced by Gualtieri (Branes on Poisson varieties, The many facets of geometry: a tribute to Nigel Hitchin (2010) (Oxford: Oxford University Press) pp. 368–395).

Almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds

As a generalization of Riemannian submersions, horizontally conformal submersions, semi-invariant submersions, h-semi-invariant submersions, almost h-semi-invariant submersions, conformal semi-invariant submersions, we introduce h-conformal semi-invariant submersions and almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We study their properties: the geometry of foliations, the conditions for total manifolds to be locally product manifolds, the conditions for such maps to be totally geodesic. Finally, we give some examples of such maps.

On the topological complexity of Grassmann manifolds

Abstract We prove that the topological complexity of a quaternionic flag manifold is half of its real dimension. For the real oriented Grassmann manifolds G͠ n,k , 3 ≤ k ≤ [n/2], the zero-divisor cup-length of the rational cohomology of G͠ n,k is computed in terms of n and k which gives a lower bound for the topological complexity of G͠ n,k , TC(G͠ n,k ). When k = 3, it is observed in certain cases that better lower bounds for TC(G͠ n,3) are obtained using ℤ2-cohomology.

Non-linear Morse–Bott functions on quaternionic Stiefel manifolds

In the Stiefel manifold Xn,k, we replace Frankel linear height function by a quadratic one. We prove this is still a Morse–Bott function, whose structure of critical levels presents a dichotomy according to the sign of n−2k. The critical submanifolds are no longer Grassmannians but total spaces of fibrations of basis a product of two Grassmannians. We explicitly integrate the gradient flow.

English

We introduce the notions of h-conformal anti-invariant submersions and h-conformal Lagrangian submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, anti-invariant submersions, h-anti-invariant submersions, h-Lagrangian submersion, conformal anti-invariant submersions. We investigate their properties: the integrability of distributions, the geometry of foliations, the conditions for such maps to be totally geodesic, etc. Finally, we give some examples of such maps.

Unscented Dual Quaternion Particle Filter for SE(3) Estimation

We present a novel dual quaternion filter for recursive estimation of rigid body motions. Based on the sequential Monte Carlo scheme, particles are deployed on the manifold of unit dual quaternions. This allows non-parametric modeling of arbitrary distributions underlying on the SE(3) group. The proposal distribution for importance sampling is estimated particle-wise by a novel dual quaternion unscented Kalman filter (DQ-UKF). It is adapted to the manifold geometric structure and drives the prior particles towards high-likelihood regions on the manifold. The resultant unscented dual quaternion particle filter (U-DQPF) incorporates the most recently observed evidence, raising the particle efficiency considerably for nonlinear pose estimation tasks. Compared with ordinary particle filters and other parametric model-based dual quaternion filtering schemes, the proposed U-DQPF shows superior performance in nonlinear SE(3) estimation.

A generalized Wintgen inequality for quaternionic CR-submanifolds

The aim of this paper is to extend the classical DDVV inequality to CR-submanifolds of quaternionic Kahler manifolds of constant quaternionic sectional curvature. We first obtain a more general inequality involving the normalized scalar normal curvature $$\rho _N$$ (defined from the second fundamental form) and then derive a DDVV-type inequality involving the normalized normal scalar curvature $$\rho ^{\perp }$$ (defined from the normal curvature tensor) for CR-submanifolds in quaternionic ambient space. We also characterize the second fundamental form of those submanifolds for which the equality case holds and give a nontrivial example of submanifold satisfying the equality case identically.