Quaternionic Projective Space Research Articles

On quaternionic bisectional curvature

AbstractIn this article we study the concept of quaternionic bisectional curvature introduced by Chow and Yang in (J Differ Geom 29(2):361–372, 1989) for quaternion-Kähler manifolds. We show that non-negative quaternionic bisectional curvature is only realized for the quaternionic projective space $$\mathbb {H}\textrm{P}^n$$ H P n . We also show that all symmetric quaternion-Kähler manifolds different from $$\mathbb {H}\textrm{P}^n$$ H P n admit quaternionic lines of negative quaternionic bisectional curvature. In particular this implies that non-negative sectional curvature does not imply non-negative quaternionic bisectional curvature. Moreover we give a new and rather short proof of a classification result by A. Gray on compact Kähler manifolds of non-negative sectional curvature.

Rational homotopy type and nilpotency of mapping spaces between Quaternionic projective spaces

The rational homotopy type of a mapping space is a way to describe the structure of the space using the algebra of its homotopy groups and the differential graded algebra of its cochains. An L∞-model is a graded Lie algebra with a family of higher-order brackets satisfying the generalized Jacobi identity and antisymmetry. It can be used to study the rational homotopy type of a space. The nilpotency index of an L∞-model is useful in understanding a space's algebraic structure. In this paper, we compute the rational homotopy type of the component of some mapping spaces between projective spaces and determine the nilpotency index of corresponding L∞-models.

Equivariant harmonic maps of the complex projective spaces into the quaternion projective spaces

We classify equivariant harmonic maps of the complex projective spaces CPm into the quaternion projective spaces. To do this, we employ differential geometry of vector bundles and connections. When the domain is the complex projective line, we have one parameter family of those maps. (This result is already shown in [2] and [4] in other ways). However, when m≧2, we will obtain the rigidity results.

Circle Actions on Oriented Manifolds With 3 Fixed Points

Abstract Let the circle group act on a compact oriented manifold $M$ with a non-empty discrete fixed point set. Then the dimension of $M$ is even. If $M$ has one fixed point, $M$ is the point. In any even dimension, such a manifold $M$ with two fixed points exists, a rotation of an even dimensional sphere. Suppose that $M$ has three fixed points. Then the dimension of $M$ is a multiple of 4. Under the assumption that each isotropy submanifold is orientable, we show that if $\dim M=8$, then the weights at the fixed points agree with those of an action on the quaternionic projective space $\mathbb{H}\mathbb{P}^{2}$, and show that there is no such 12-dimensional manifold $M$.

The string topology coproduct on complex and quaternionic projective space

On the free loop space of compact symmetric spaces Ziller introduced explicit cycles generating the homology of the free loop space. We use these explicit cycles to compute the string topology coproduct on complex and quaternionic projective space. The behavior of the Goresky-Hingston product for these spaces then follows directly.

Real hypersurfaces with Ricci–Bourguignon soliton in the complex two-plane Grassmannians

The study of Ricci-Bourguignon soliton on real hypersurfaces in the complex two-plane Grassmannian [Formula: see text] is first investigated. It is proved that there exists a shrinking Ricci-Bourguignon soliton on a Hopf real hypersurface [Formula: see text] in [Formula: see text] by using pseudo-anticommuting Ricci tensor. Moreover, we have proved that there does not exist a nontrivial gradient Ricci-Bourguignon soliton ([Formula: see text]) on real hypersurfaces with isometric Reeb flow in the complex two-plane Grassmannian [Formula: see text]. Among the class of contact hypersurface in [Formula: see text], we also prove that there does not exist a nontrivial gradient Ricci-Bourguignon in [Formula: see text] over the totally geodesic and totally real quaternionic projective space [Formula: see text] in [Formula: see text], [Formula: see text].

Higher derivative couplings of hypermultiplets

We construct the four-derivative supersymmetric extension of (1, 0), 6D supergravity coupled to Yang-Mills and hypermultiplets. The hypermultiplet scalars are taken to parametrize the quaternionic projective space Hp(n) = Sp(n, 1)/Sp(n) × Sp(1)R. The hyperscalar kinetic term is not deformed, and the quaternionic Kähler structure and symmetries of Hp(n) are preserved. The result is a three parameter Lagrangian supersymmetric up to first order in these parameters. Considering the case of Hp(1) we compare our result with that obtained from the compactification of 10D heterotic supergravity on four-torus, consistently truncated to N = (1, 0), in which the hyperscalars parametrize SO(1, 4)/SO(4). We find that depending on how the Sp(1) is embedded in the SO(4), the results agree for a specific value of the parameter that governs the higher derivative hypermultiplet couplings.

Simultaneous linearization of diffeomorphisms of isotropic manifolds

Suppose that M is a closed isotropic Riemannian manifold and that R_1,\dots ,R_m generate the isometry group of M . Let f_1,\dots ,f_m be smooth perturbations of these isometries. We show that the f_i are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian [Duke Math. J. {136}, 475–505 (2007)] from S^n to real, complex, and quaternionic projective spaces. In addition, we identify and remedy an oversight in that earlier work.

Minimal δ(2)-ideal Lagrangian submanifolds and the quaternionic projective space

We construct an explicit map from a generic minimal δ(2)-ideal Lagrangian submanifold of Cn to the quaternionic projective space HPn−1, whose image is either a point or a minimal totally complex surface. A stronger result is obtained for n=3, since the above mentioned map then provides a one-to-one correspondence between minimal δ(2)-ideal Lagrangian submanifolds of C3 and minimal totally complex surfaces in HP2 which are moreover anti-symmetric. Finally, we also show that there is a one-to-one correspondence between such surfaces in HP2 and minimal Lagrangian surfaces in CP2.

Limit sets of cyclic quaternionic Kleinian groups

In this paper, we consider the natural action of $$\textrm{SL}(3, \mathbb {H})$$ on the quaternionic projective space $$ {\mathbb {P}}_{\mathbb {H}}^2$$ . Under this action, we investigate limit sets for cyclic subgroups of $$\textrm{SL}(3, \mathbb {H})$$ . We compute two types of limit sets, which were introduced by Kulkarni and Conze-Guivarc’h, respectively.

On the space of riemannian metrics satisfying surgery stable curvature conditions

We utilize a condition for algebraic curvature operators called surgery stability as suggested by the work of Hoelzel to investigate the space of riemannian metrics over closed manifolds satisfying these conditions. Our main result is a parametrized Gromov–Lawson construction with not necessarily trivial normal bundles and shows that the homotopy type of this space of metrics is invariant under surgeries of a suitable codimension. This is a generalization of a well-known theorem by Chernysh and Walsh for metrics of positive scalar curvature. As an application of our method, we show that the space of metrics of positive scalar curvature on quaternionic projective spaces are homotopy equivalent to that on spheres.

Geodesic complexity via fibered decompositions of cut loci

The geodesic complexity of a Riemannian manifold is a numerical isometry invariant that is determined by the structure of its cut loci. In this article we study decompositions of cut loci over whose components the tangent cut loci fiber in a convenient way. We establish a new upper bound for geodesic complexity in terms of such decompositions. As an application, we obtain estimates for the geodesic complexity of certain classes of homogeneous manifolds. In particular, we compute the geodesic complexity of complex and quaternionic projective spaces with their standard symmetric metrics.

The moduli space of points in quaternionic projective space

Let Mn,m;Fℙn be the configuration space of m-tuples of pairwise distinct points in Fℙn, that is, the quotient of the set of m-tuples of pairwise distinct points in Fℙn with respect to the diagonal action of PU(1,n;F) equipped with the quotient topology. In this paper, by mainly using the rotation-normalized and the block-normalized algorithms, we construct the parameter spaces of both Mn,m;∂Hℍn and Mn,m;ℙ(V+), respectively.

Stable Submanifolds in the Product of Projective Spaces

We provide a classification theorem for compact stable minimal immersions (CSMI) of codimension 1 or dimension 1 (codimension 1 and 2 or dimension 1 and 2) in the product of a complex (quaternionic) projective space with any other Riemannian manifold. We characterize the complex minimal immersions of codimension 2 or dimension 2 as the only CSMI in the product of two complex projective spaces. As an application, we characterize the CSMI of codimension 1 or dimension 1 (codimension 1 and 2 or dimension 1 and 2) in the product of a complex (quaternionic) projective space with any compact rank one symmetric space.

Symmetries, tensors, and the Horrocks bundle

A study of tensors on the quaternionic projective plane HP2 arising from a stable 3-form on C6 and an associated action of SU(3) is related to the existence of a holomorphic rank 3 vector bundle over CP5 discovered by Horrocks. It also leads to the construction of SU(3) invariant Spin(7) structures on HP2, which are characterised in terms of associated 4-forms.

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.

Stability of Compact Symmetric Spaces

In this article, we study the stability problem for the Einstein–Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover, we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.

Inertia groups and smooth structures on quaternionic projective spaces

Abstract This paper deals with certain results on the number of smooth structures on quaternionic projective spaces, obtained through the computation of inertia groups and their analogues, which in turn are computed using techniques from stable homotopy theory. We show that the concordance inertia group is trivial in dimension 20, but there are many examples in high dimensions where the concordance inertia group is non-trivial. We extend these to computations of concordance classes of smooth structures. These have applications to 3-sphere actions on homotopy spheres and tangential homotopy structures.

Explicit formulas of the heat kernel on the quaternionic projective spaces

We consider the heat equation on the quaternionic projective space $$$P_n(ℍ)$$$, and we establish two formulas for the heat kernel, a series expansion involving Jacobi polynomials, and an integral (...)

Cohomology classification of spaces with free S1 and S3-actions

This paper gives the cohomology classification of finitistic spaces X equipped with free actions of the group G = S3 and the cohomology ring of the orbit space X/G is isomorphic to the integral cohomology quaternion projective space HPn. We have proved that the integral cohomology ring of X is isomorphic either to S4n+3 or S3 ? HPn. Similar results with other coefficient groups and for G = S1 actions are also discussed. As an application, we determine a bound of the index and co-index of cohomology sphere S2n+1 (resp. S4n+3) with respect to S1-actions (resp. S3-actions).