Quaternionic Projective Space Research Articles

Polar foliations on quaternionic projective spaces

We classify irreducible polar foliations of codimension q on quaternionic projective spaces ℍPn, for all (n, q) ≠ (7,1). We prove that all irreducible polar foliations of any codimension (resp. of codimension one) on ℍPn are homogeneous if and only if n + 1 is a prime number (resp. n is even or n = 1). This shows the existence of inhomogeneous examples of codimension one and higher.

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 THE SYMMETRIC SQUARES OF COMPLEX AND QUATERNIONIC PROJECTIVE SPACE

AbstractThe problem of computing the integral cohomology ring of the symmetric square of a topological space has long been of interest, but limited progress has been made on the general case until recently. We offer a solution for the complex and quaternionic projective spaces$\mathbb{K}$Pn, by utilising their rich geometrical structure. Our description involves generators and relations, and our methods entail ideas from the literature of quantum chemistry, theoretical physics, and combinatorics. We begin with the case$\mathbb{K}$P∞, and then identify the truncation required for passage to finiten. The calculations rely upon a ladder of long exact cohomology sequences, which compares cofibrations associated to the diagonals of the symmetric square and the corresponding Borel construction. These incorporate the one-point compactifications of classic configuration spaces of unordered pairs of points in$\mathbb{K}$Pn, which are identified as Thom spaces by combining Löwdin's symmetric orthogonalisation (and its quaternionic analogue) with a dash ofPingeometry. The relations in the ensuing cohomology rings are conveniently expressed using generalised Fibonacci polynomials. Our conclusions are compatible with those of Gugnin mod torsion and Nakaoka mod 2, and with homological results of Milgram.

SUBMAXIMALLY SYMMETRIC ALMOST QUATERNIONIC STRUCTURES

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 ℍP n , which is at and has the symmetry algebra $$ \mathfrak{sl}\left(n+1,\mathrm{\mathbb{H}}\right) $$ of dimension 4n 2 + 8n + 3. For non-flat almost quaternionic manifolds we compute the next biggest (submaximal) symmetry dimension. We show that it is equal to 4n 2 –4n+9 for n > 1 (it is equal to 8 for n = 1). This is realized both by a quaternionic structure (torsion–free) and by an almost quaternionic structure with vanishing quaternionic Weyl curvature.

Classification of homogeneous minimal immersions from $$S^2$$ S 2 to $${\mathbb {H}}P^n$$ H P n

In this paper we determine all homogeneous minimal immersions of 2-spheres in quaternionic projective spaces \({\mathbb {H}}P^n\).

An inclusive immersion into a quaternionic manifold and its invariants

We introduce a quaternionic invariant for an inclusive immersion into a quaternionic manifold, which is a quaternionic object corresponding to the Willmore functional. The lower bound of this invariant is given by topological invariant and the equality case can be characterized in terms of the natural twistor lift. When the ambient manifold is the quaternionic projective space and the natural twistor lift is holomorphic, we obtain a relation between the quaternionic invariant and the degree of the image of the natural twistor lift as an algebraic curve. Moreover the first variation formula for the invariant is obtained. As an application of the formula, if the natural twistor lift is a harmonic section, then the surface is a stationary point under any variations such that the induced complex structures do not vary.

Periods of continuous maps on some compact spaces

The objective of this paper is to provide information on the set of periodic points of a continuous self-map defined in the following compact spaces: (the n-dimensional sphere), (the product space of the n-dimensional with the m-dimensional spheres), (the n-dimensional complex projective space) and (the n-dimensional quaternion projective space). We use as main tool the action of the map on the homology groups of these compact spaces.

Solvable stochastic differential games in rank one compact symmetric spaces

ABSTRACTSome nonlinear stochastic differential games are formulated in the family of complex and quaternion projective spaces that are among the rank one compact symmetric spaces that consist of the spheres, the projective spaces over , and and one arising from an exceptional Lie algebra called the Cayley plane. The payoff functionals for the differential games are obtained from some eigenfunctions of the radial part of the Laplacians for these Riemannian manifolds and these payoffs induce symmetries for the game problems that reduce the required analysis to a radial direction in these manifolds. These projective spaces are given a natural Riemannian metric from the Killing forms of the compact Lie groups for these spaces that are the special unitary groups, SU(n), and the symplectic groups, Sp(n). Explicit optimal control strategies are obtained for these differential games and the explicit payoffs are given. A countable family of distinct solvable stochastic differential games can be obtained for each of these compact symmetric spaces.

A minimal triangulation of the quaternionic projective plane

A minimal triangulation of the quaternionic projective plane

Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes

We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes to a large class of symmetric ambient spaces the estimates obtained in the previous works on the mean curvature flow of hypersurfaces in Euclidean space and in the sphere.

Almost CR quaternionic manifolds and their immersibility in $$\mathbb {H}$$ H P $$^n$$ n .

We apply the general theory of codimension one integrability conditions for G-structures developed in Santi (Ann Mat Pura Appl 195:1463–1489, 2016) to the case of quaternionic CR geometry. We obtain necessary and sufficient conditions for an almost CR quaternionic manifold to admit local immersions as an hypersurface of the quaternionic projective space. We construct a deformation of the standard quaternionic contact structure on the quaternionic Heisenberg group which does not admit local immersions in any quaternionic manifold.

Characterizing the unit ball by its projective automorphism group

In this paper we study the projective automorphism group of domains in real, complex, and quaternionic projective space and present two new characterizations of the unit ball in terms of the size of the automorphism group and the regularity of the boundary.

On submersion and immersion submanifolds of a quaternionic projective space

Abstract We study submanifolds of a quaternionic projective space, it is of great interest how to pull down some formulae deduced for submanifolds of a sphere to those for submanifolds of a quaternionic projective space.

Optimal simplices and codes in projective spaces

We find many tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence (and abundance) of several hitherto unknown families of simplices in quaternionic projective spaces and the octonionic projective plane. The most noteworthy cases are 15-point simplices in HP^2 and 27-point simplices in OP^2, both of which are the largest simplices and the smallest 2-designs possible in their respective spaces. These codes are all universally optimal, by a theorem of Cohn and Kumar. We also show the existence of several positive-dimensional families of simplices in the Grassmannians of subspaces of R^n with n <= 8; close numerical approximations to these families had been found by Conway, Hardin, and Sloane, but no proof of existence was known. Our existence proofs are computer-assisted, and the main tool is a variant of the Newton-Kantorovich theorem. This effective implicit function theorem shows, in favorable conditions, that every approximate solution to a set of polynomial equations has a nearby exact solution. Finally, we also exhibit a few explicit codes, including a configuration of 39 points in OP^2 that form a maximal system of mutually unbiased bases. This is the last tight code in OP^2 whose existence had been previously conjectured but not resolved.

Quaternionic contact hypersurfaces in hyper-Kähler manifolds

We describe explicitly all quaternionic contact hypersurfaces (qc-hypersurfaces) in the flat quaternion space $$\mathbb {H}^{n+1}$$ and the quaternion projective space. We show that up to a quaternionic affine transformation a qc-hypersurface in $$\mathbb {H}^{n+1}$$ is contained in one of the three qc-hyperquadrics in $$\mathbb {H}^{n+1}$$ . Moreover, we show that an embedded qc-hypersurface in a hyper-Kahler manifold is qc-conformal to a qc-Einstein space and the Riemannian curvature tensor of the ambient hyper-Kahler metric is degenerate along the hypersurface.

Isometric immersions from a Kähler manifold into the quaternionic projective space

In this paper we present a rigidity theorem for isometric immersions from Kahler manifolds into a quaternionic projective space (or its non compact real form), when its nullity index is everywhere positive.

A minimal triangulation of the quaternionic projective plane

A minimal triangulation of the quaternionic projective plane

Delaunay-Type Hypersurfaces in Cohomogeneity One Manifolds

Classical Delaunay surfaces are highly symmetric constant mean curvature (CMC) submanifolds of space forms. We prove the existence of Delaunay-type hypersurfaces in a large class of compact manifolds, using the geometry of cohomogeneity one group actions and variational bifurcation techniques. Our construction specializes to the classical examples in round spheres, and allows to obtain Delaunay-type hypersurfaces in many other ambient spaces, ranging from complex and quaternionic projective spaces to Kervaire exotic spheres.

On conformal minimal immersions of constant curvature from $$S^2$$ S 2 to $${ HP }^n$$ H P n

We generalize here our general characterization of the harmonic sequence generated by conformal minimal immersions from $$S^2$$ to the quaternionic projective space $${ HP }^n$$ . Then we give a classification theorem of linearly full unramified conformal minimal immersions of constant curvature from $$S^2$$ to $${ HP }^n$$ under some conditions.

Non-Kähler expanding Ricci solitons, Einstein metrics, and exotic cone structures

We produce new non-K\"ahler, non-Einstein, complete expanding gradient Ricci solitons with conical asymptotics and underlying manifold of the form $\R^2 \times M_2 \times \cdots \times M_r$, where $r \geq 2$ and $M_i$ are arbitrary closed Einstein spaces with positive scalar curvature. We also find numerical evidence for complete expanding solitons on the vector bundles whose sphere bundles are the twistor or ${\rm Sp}(1)$ bundles over quaternionic projective space.