We consider the moduli space M_r of polygons with fixed side lengths in five-dimensional eucledian space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between M_r and a weighted quotient of the n-fold product of the quaternionic projective line HP^1 by the diagonal PSL(2,H)-action. We explore the relation between M_r and the fixed point set of an anti-symplectic involution on a GIT quotient Gr(2,4)^n/SL(4,C). We generalize the Gel'fand-MacPherson correspondence to more general complex Grassmannians and to the quaternionic context, and realize our space M_r as a quotient of a subspace in the quaternionic Grassmannian Gr_H(2,n) by the action of the group Sp(1)^n. We also give analogues of the Gel'fand-Tsetlin coordinates on the space of quaternionic Hermitean matrices and briefly describe generalized action-angle coordinates on M_r.
We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Knutson and Hausmann using the Hopf map on quaternions, from the complex Stiefel manifold of 2-frames in n-space to the space of closed n-gons in 3-space of total length 2. Our probability measure on polygon space is defined by pushing forward Haar measure on the Stiefel manifold by this map. A similar construction yields a probability measure on plane polygons which comes from a real Stiefel manifold. The edgelengths of polygons sampled according to our measures obey beta distributio…
The purpose of this paper is to describe certain natural 4-vector fields on quaternionic flag manifolds, which geometrically determine the Bruhat cell decomposition. This structure naturally descends from the symplectic group, where it is related to the dressing action given by the Iwasawa decomposition of the general linear group over the quaternions.
The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean c…
Modelled on a real hypersurface in a quaternionic manifold, we introduce a quaternionic analogue of CR structure, called quaternionic CR structure. We define the strong pseudoconvexity of this structure as well as the notion of quaternionic pseudohermitian structure. Following the construction of the Tanaka-Webster connection in complex CR geometry, we construct a canonical connection associated with a quaternionic pseudohermitian structure, when the underlying quaternionic CR structure satisfies the ultra-pseudoconvexity which is stronger than the strong pseudoconvexity. Comparison to Biquard…
The fibre bundles adjoint to generalized almost quaternionic structures are studied. The most important classes of generalized almost quaternionic manifolds are considered.
We study the Penrose transform for the `quaternionic objects' whose twistor spaces are complex manifolds endowed with locally complete families of embedded Riemann spheres with positive normal bundles.
This paper describes the foundations of a differential geometry of a quaternionic curves. The Frenet-Serret equations and the evolutes and evolvents of a particular quaternionic curve are accordingly determined. This new formulation takes benefit of the quaternionic structure and the results are much simpler than the present formulations of quaternionic curves.
This is the first comprehensive introduction to the authors' recent attempts toward a better understanding of the global concepts behind spinor representations of surfaces in 3-space. The important new aspect is a quaternionic-valued function theory, whose "meromorphic functions" are conformal maps into quaternions, which extends the classical complex function theory on Riemann surfaces. The first results along these lines were presented at the ICM 98 in Berlin. An important new invariant of the quaternionic holomorphic theory is the Willmore energy. For quaternionic holomorphic curves in the …
We describe the 8-dimensional Wolf spaces as cohomogeneity one SU(3)-manifolds, and discover perturbations of the quaternion-kaehler metric on the simply-connected 8-manifold G_2/SO(4) that carry a closed fundamental 4-form but are not Einstein.
Notions of self-dual and anti self-dual almost quaternionic structures are introduced. The complete classification of self-dual and anti self-dual generalized Kaehler manifolds is obtained.
Motivated by the analogies between the projective and the almost quaternionic geometries, we study the generalized planar curves and mappings. We follow, recover, and extend the classical approach as developed by Mikes and Sinyukov. Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of H--planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaterni…
In this paper we introduce a new algebraic device, which enables us to treat the quaternions as though they were a commutative field. This is of interest both for its own sake, and because it can be applied to develop an "algebraic geometry" of noncompact hypercomplex manifolds. The basic building blocks of the theory are AH modules, which should be thought of "vector spaces" over the quaternions. An AH-module is a left module over the quaternions H, together with a real vector subspace. There are natural concepts of linear map and tensor product of AH-modules, which have many of the propertie…
Almost para-quaternionic structures on smooth manifolds of dimension $2n$ are equivalent to almost Grassmannian structures of type $(2,n)$. We remind the equivalence and exhibit some interrelations between subjects that were previously studied independently from the para-quaternionic and the Grassmannian point of view. In particular, we relate the respective normalization conditions, distinguished curves and twistor constructions.
We study hyperkahler cones and their corresponding quaternion-Kahler spaces. We present a classification of 4(n-1)-dimensional quaternion-Kahler spaces with n abelian quaternionic isometries, based on dualizing superconformal tensor multiplets. These manifolds characterize the geometry of the hypermultiplet sector of perturbative moduli spaces of type-II strings compactified on a Calabi-Yau manifold. As an example of our construction, we study the universal hypermultiplet in detail, and give three inequivalent tensor multiplet descriptions. We also comment on the construction of quaternion-Kah…
We establish quaternionic contact (qc) versions of the so called Almost Schur Lemma, which give estimations of the qc scalar curvature on a compact qc manifold to be a constant in terms of the norm of the $[-1]$-component and the norm of the trace-free part of the $[3]$-component of the horizontal qc Ricci tensor and the torsion endomorphism, under certain positivity conditions.
In this article we discuss the interaction between the geometry of a quaternion-Kahler manifold M and that of the Grassmannian G(3,g) of oriented 3-dimensional subspaces of a compact Lie algebra g. This interplay is described mainly through the moment mapping induced by the action of a group G of quaternionic isometries on M. We give an alternative expression for the endomorphisms I_{1},I_{2},I_{3}, both in terms of the holonomy representation of M and the structure of the Grassmannian's tangent space. A correspondence between the solutions of respective twistor-type equations on M and G(3,g) …
An important problem in quaternionic hyperbolic geometry is to classify ordered $m$-tuples of pairwise distinct points in the closure of quaternionic hyperbolic n-space, $\overline{{\bf H}_\bh^n}$, up to congruence in the holomorphic isometry group ${\rm PSp}(n,1)$ of ${\bf H}_\bh^n$. In this paper we concentrate on two cases: $m=3$ in $\overline{{\bf H}_\bh^n}$ and $m=4$ on $\partial{\bf H}_\bh^n$ for $n\geq 2$. New geometric invariants and several distance formulas in quaternionic hyperbolic geometry are introduced and studied for this problem. The congruence classes are completely described…
We provide a general criteria for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P) of dimension bigger or equal to eight, in terms of the integrability of two or three sections of the defining rank three vector bundle P. We relate it with the integrability of the canonical almost complex structure of the twistor space and to the integrability of the canonical almost para-complex structure of the reflector space of (M,P). We show that (M, P) has plenty of locally defined, compatible, complex and para-complex structures, provided that P is…
The generalized Feix--Kaledin construction shows that c-projective $2n$-manifolds with curvature of type $(1,1)$ are precisely the submanifolds of quaternionic $4n$-manifolds which are fixed points set of a special type of quaternionic $S^1$ action $v$. In this paper, we consider this construction in the presence of infinitesimal symmetries of the two geometries. First, we prove that the submaximally symmetric c-projective model with type $(1,1)$ curvature is a submanifold of a submaximally symmetric quaternionic model, and show how this fits into the construction. We give conditions for when …
The moduli space of the Calabi-Yau three-folds, which play a role as superstring ground states, exhibits the same {\em special geometry} that is known from nonlinear sigma models in $N=2$ supergravity theories. We discuss the symmetry structure of special real, complex and quaternionic spaces. Maps between these spaces are implemented via dimensional reduction. We analyze the emergence of {\em extra} and {\em hidden} symmetries. This analysis is then applied to homogeneous special spaces and the implications for the classification of homogeneous quaternionic spaces are discussed.
The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study geometry of closed polygonal lines in $\bbRP^d$ and prove that polygons satisfying a certain convexity condition have at least d+1 flattenings. This result provides a new approach to the above mentioned classical theorems.
The Alesker-Poincare pairing for smooth valuations on manifolds is expressed in terms of the Rumin differential operator acting on the cosphere-bundle. It is shown that the derivation operator, the signature operator and the Laplace operator acting on smooth valuations are formally self-adjoint with respect to this pairing. As an application, the product structure of the space of SU(2)- and translation invariant valuations on the quaternionic line is described. The principal kinematic formula on the quaternionic line is stated and proved.
We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely integrable systems on these moduli spaces by bending n-gons along their diagonals. The results of this paper are the analogues of the results of the first two authors for n-gon linkages in Euclidean 3-space in JDG 44. We conclude by proving by a deformation argument that the moduli spaces of n-gon linkages in hyperbolic 3-space and Euclidean 3-space with the same…
Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the `quaternionic-like manifolds'. These contain, as particular subclasses, the CR quaternionic and the $ρ$-quaternionic manifolds. Moreover, the notion of `heaven space' finds its adequate level of generality in this setting: (essentially) any real analytic quaternionic-like manifold admits a (germ) unique heaven space, which is a $ρ$-quaternionic manifold. We, also, give a natural construction of homogeneous complex …
Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine invariants, equal-volume polygons can also be used to estimate projective invariants of a planar curve. This theory has many potential applications, among them evaluation of the quality and computation of affine invariants of silhouette curves.
We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between conformal hypercomplex manifolds (i.e. those that have a closed homothetic Killing vector) and quaternionic manifolds of one quaternionic dimension less. An important role is played by 'ξ-transformations', relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds. In this map, the subclass of conformal hyper-Kae…
Following the Cartans's original method of equivalence supported by methods of parabolic geometry, we provide a complete solution for the equivalence problem of quaternionic contact structures, that is, the problem of finding a complete system of differential invariants for two quaternionic contact manifolds to be locally diffeomorphic. This includes an explicit construction of the corresponding Cartan geometry and detailed information on all curvature components.
Let Z be a compact complex (2n+1)-manifold which carries a {\em complex contact structure}, meaning a codimension-1 holomorphic sub-bundle D of TZ which is maximally non-integrable. If Z admits a Kähler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kähler manifold (M^{4n}, g). If Z also admits a second complex contact structure, then Z= CP_{2n+1}. As an application, we give several new characterizations of the Riemannian manifold HP_n=Sp(n+1)/( Sp(n)\times Sp(1)).
We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kaehler manifold (K,Γ) as the rank of a natural quotient of Γ, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kaehler reduction of compact Vaisman manifolds produces Vaisman manifolds and…
The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n).Sp(1), QKT-connection. We study the geometry of QKT-connections. We find conditions to the existence of a QKT-connection and prove that if it exists it is unique. Studying conformal transformations we obtain a lot of (compact) examples of QKT manifolds. We present a (local) description of 4-dimensional homogeneous QKT structures relying on the known result of naturally reductive homogeneous Riemannian manifolds. We con…
Following a recent paper by Baryshnikov and Zharnitskii, we consider outer billiards in the plane possessing invariant curves consisting of periodic orbits. We prove the existence and abundance of such tables using tools from sub-Riemannian geometry. We also prove that the set of 3-periodic outer billiard orbits has zero measure.
A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern-Moser tensor in CR geometry. It is shown that a quaternionic contact manifold is locally quaternionic contact conformal to the standard flat quaternionic contact structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-sasakian structure o…
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 o…
A hypercomplex manifold is a manifold equipped with a triple of complex structures $I, J, K$ satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics, and prove a quaternionic analogue of A.D. Aleksandrov and Chern-Levine-Nirenberg theorems.
In this note, we prove that the CR manifold which is induced from the canonical parabolic geometry of a quaternionic contact (qc) manifold via a Fefferman-type construction is equivalent to the CR twistor space of the qc manifold defined by O. Biquard.
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 classification of polar actions on Hermitian symmetric spaces. In the second part we prove that polar actions on Wolf spaces are quaternion-coisotropic and that isometric actions on these spaces admit an orbit of special type, analogous to the existence of a complex orbit for an isometric act…
We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\mathfrak{p}$ in the infinitesimal isotropy representation $\mathbb{W}$ of a larger self-dual symmetric R-space $H\cdot\mathfrak{q}$. We also give a classification of projective parabolic geometries with $H\cdot\mathfrak{q}$ irreducible which, in addition to the aforementioned class…
This is a survey on quaternion Hermitian Weyl (locally conformally quaternion Kähler) and hyperhermitian Weyl (locally conformally hyperkähler) manifolds. These geometries appear by requesting the compatibility of some quaternion Hermitian or hyperhermitian structure with a Weyl structure. The motivation for such a study is two-fold: it comes from the constantly growing interest in Weyl (and Einstein-Weyl) geometry and, on the other hand, from the necessity of understanding the existing classes of quaternion Hermitian manifolds.
We study the moduli space of quaternionic Kaehler structures on a compact manifold of dimension 4n (n>2) from a point of view of Riemannian geometry, not twistor theory. Then we obtain a rigidity theorem for quaternionic Kaehler structures of nonzero scalar curvature by observing the moduli space.
We study the quaternionic Calabi-Yau problem in HKT geometry introduced by Alesker and Verbitsky on 8-dimensional 2-step nilmanifolds with an abelian hypercomplex structure. We show that the quaternionic Monge-Ampère equation on these manifolds can always be solved for every data which is invariant by the action of a 3-dimensional torus.
We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in seven dimensions, when this condition is equivalent to integrability in the sense of Duchemin. We prove that the choice of a reduction to Sp(n)H^* (or equivalently, a complement of the qc distribution) yields a unique K-connection satisfying natural conditions on torsion and curvat…
We discuss the geometry of the c-map from projective special Kähler to quaternionic Kähler manifolds using the twist construction to provide a global approach to Hitchin's description. As found by Alexandrov et al. and Alekseevsky et al. this is related to the quaternionic flip of Haydys. We prove uniqueness statements for several steps of the construction. In particular, we show that given a hyperKähler manifold with a rotating symmetry, there is essentially only a one parameter degree of freedom in constructing a quaternionic Kähler manifold of the same dimension. We demonstrate how examples…
We study the symplectic geometry of the moduli spaces $M_r=M_r(\s^3)$ of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of $n$ conjugacy classes in SU(2), denoted $C_r^n$, by the diagonal conjugation action of SU(2). Here $C_r^n$ is a quasi-Hamiltonian SU(2)-space. An integrable Hamiltonian system is constructed on $M_r$ in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on $…
Any oriented 4-dimensional real vector bundle is naturally a line bundle over a bundle of quaternion algebras. In this paper we give an account of modules over bundles of quaternion algebras, discussing Morita equivalence, characteristic classes and K-theory. The results have been used to describe obstructions for the existence of almost quaternionic structures on 8-dimensional Spinc manifolds and may be of some interest, also, in quaternionic and algebraic geometry.
We give an overview of some recent results in hypersymplectic and para-quaternionic Kahler geometry, and introduce the notion of split three-Sasakian manifold. In particular, we discuss the twistor spaces and Swann bundles of para-quaternionic Kahler manifolds. These are used to classify examples with a fully homogeneous action of a semi-simple Lie group, and to construct distinct para-quaternionic Kahler metrics from indefinite real analytic conformal manifolds. We also indicate how the theory of toric varieties gives rise to constructions of hypersymplectic manifolds.
We consider timelike and spacelike reductions of 4D, N = 2 Minkowskian and Euclidean vector multiplets coupled to supergravity and the maps induced on the scalar geometry. In particular, we investigate (i) the (standard) spatial c-map, (ii) the temporal c-map, which corresponds to the reduction of the Minkowskian theory over time, and (iii) the Euclidean c-map, which corresponds to the reduction of the Euclidean theory over space. In the last two cases we prove that the target manifold is para-quaternionic Kahler. In cases (i) and (ii) we construct two integrable complex structures on the targ…
It has long been known that differential forms on complex manifolds can be decomposed under the action of the complex structure to give the Dolbeault complex. This paper presents an analogous double complex for quaternionic manifolds using the fact that the cotangent space is isomorphic to a quaternionic vector space. This defines an action of the group Sp(1) of unit quaternions on the cotangent space, which induces an action of Sp(1) on the space of k-forms. A double complex is obtained by decomposing the k-forms into irreducible representations of Sp(1), resulting in new 'quaternionic Dolbea…
