Quaternionic Projective Space Research Articles

$\mathcal{C}^{1}$ SELF-MAPS ON $\mathbb{S}^{n}$, $\mathbb{S}^{n}\times \mathbb{S}^{m}$, $\mathbb{C}$P$^{n}$ AND $\mathbb{H}$P$^{n}$ WITH ALL THEIR PERIODIC ORBITS HYPERBOLIC

We study in its homological class the periodic structure of the $\mathcal{C}^{1}$ self--maps on the manifolds $\mathbb{S}^{n}$ (the $n$--dimensional sphere), $\mathbb{S}^{n}\times \mathbb{S}^{m}$ (the product space of the $n$--dimensional with the $m$--dimensional spheres), $\mathbb{C}$P$^{n}$ (the $n$--dimensional complex projective space) and $\mathbb{H}$P$^{n}$ (the $n$--dimensional quaternion projective space), having all their periodic orbits hyperbolic.

On the Lefschetz periodic point free continuous self-maps on connected compact manifolds

We characterize the Lefschetz periodic point free self-continuous maps on the following connected compact manifolds: C P n the n-dimensional complex projective space, H P n the n-dimensional quaternion projective space, S n the n-dimensional sphere and S p × S q the product space of the p-dimensional with the q-dimensional spheres.

Universal inequalities for the eigenvalues of the biharmonic operator on submanifolds

We establish universal inequalities for the eigenvalues of the clamped plate problem on compact submanifolds of Euclidean space, of spheres and of real, complex and quaternionic projective spaces. We prove similar results for the biharmonic operator on domains of Riemannian manifolds that admit spherical eigenmaps (this includes compact homogeneous Riemannian spaces) and finally on domains of hyperbolic space.

Ricci tensor of slant submanifolds in a quaternion projective space

We study slant submanifolds of quaternion Kaehler manifold and in particular of a quaternion projective space. We obtain a sharp estimate of the Ricci tensor of a slant submanifold M in a quaternion projective space QP m ( 4 c ) in terms of the main extrinsic invariant, namely the square mean curvature.

Scalar curvature of QR-submanifolds with maximal QR-dimension in a quaternionic projective space

In this paper we derive an integral formula on an n-dimensional, compact, minimal QR-submanifoldM of (p−1) QR-dimension immersed in a quaternionic projective space QP(n+p)/4. Using this integral formula, we give a sufficient condition concerning with the scalar curvature of M in order that such a submanifold M is to be a tube over a quaternionic projective space.

Extrinsic eigenvalue estimates of Dirac operators on Riemannian manifolds

AbstractFor eigenvalues of generalized Dirac operators on compact Riemannian manifolds, we obtain a general inequality. By using this inequality, we study eigenvalues of generalized Dirac operators on compact submanifolds of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces. We obtain explicit bounds for the (k + 1)‐th eigenvalue of generalized Dirac operators on such objects in terms of its first k eigenvalues, which depend on the mean curvature of the embedding and the curvature term in the Bochner‐Weitzenböck formula for the square of the Dirac operator. These inequalities of eigenvalues extend the recent results in 15. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Scalar-flat Kähler orbifolds via quaternionic-complex reduction

We prove that any asymptotically locally Euclidean scalar-flat Kahler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a k-dimensional quaternionic vector space by a (k−1)-torus. In order to do so, we first prove that any compact anti-self-dual 4-orbifold with positive Euler characteristic whose isometry group contains a 2-torus is conformally equivalent, up to an orbifold covering, to a quaternionic quotient of k-dimensional quaternionic projective space by a (k − 1)-torus.

Positive Quaternion Kähler Manifolds with fourth Betti number equal to one

Positive Quaternion Kähler Manifolds are Riemannian manifolds with holonomy contained in Sp ( n ) Sp ( 1 ) and with positive scalar curvature. Conjecturally, they are symmetric spaces. In this article we are mainly concerned with Positive Quaternion Kähler Manifolds M satisfying b 4 ( M ) = 1 . Generalising a result of Galicki and Salamon we prove that M 4 n in this case is homothetic to a quaternionic projective space if 2 ≠ n ⩽ 6 .

Twistorial maps between quaternionic manifolds

We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one: • A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic. • A map between quaternionic manifolds endowed with the nonintegrable almost twistorial structures is twistorial if and only if it is quaternionic and totally- geodesic. As an application, we describe all the quaternionic maps between open sets of quaternionic projective spaces.

Quaternionic soliton equations from Hamiltonian curve flows in

A bi-Hamiltonian hierarchy of quaternion soliton equations is derived from geometric non-stretching flows of curves in the quaternionic projective space . The derivation adapts the method and results in recent work by one of us on the Hamiltonian structure of non-stretching curve flows in Riemannian symmetric spaces M = G/H by viewing as a symmetric space in terms of compact real symplectic groups and quaternion unitary groups. As main results, scalar–vector (multi-component) versions of the sine-Gordon (SG) equation and the modified Korteweg-de Vries (mKdV) equation are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of quaternionic symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map.

Maximal totally complex submanifolds of ℍℙ n : homogeneity and normal holonomy

We prove that a maximal totally complex submanifold $N^{2n}$ of the quaternionic projective space $\mathbb{H}\mathbb{P}^n$ ($n\geq 2$) is a parallel submanifold, provided one of the following conditions is satisfied: (1) $N$ is the orbit of a compact Lie group of isometries, (2) the restricted normal holonomy is a proper subgroup of ${\rm U}(n)$.

A characterization of quaternionic projective space by the conformal-Killing equation

We prove that any compact, quaternionic-Kähler manifold of dimension 4n ⩾ 8 admitting a conformal-Killing 2-form which is not Killing, is isomorphic to the quaternionic projective space, with its standard quaternionic-Kähler structure.

ADHM construction of super Yang–Mills instantons

In this paper we study the relationship between N -supersymmetric Yang–Mills instantons on the anti-chiral quaternionic super projective space H P 1 | N 2 and certain holomorphic super vector bundles on the complex super projective space P 3 | N . This is the so-called super ADHM construction. We prove mathematically exactly the super ADHM construction which has been given by physicists. In particular, we consider the monad theory of P 3 | N which generalized the Barth–Hulek construction.

Collective branch regularization of simultaneous binary collisions in the 3D N-body problem

In this work we study simultaneous binary collision (SBC) singularities of M⩽N∕2 binaries in the three dimensional classical gravitational N-body problem. We show the following: (1) In the generalized Kustaanheimo–Stiefel variables, the totality of SBC orbits, the totality of simultaneous binary collisions (SBE) orbits, and the collision singularity itself together form a real analytic submanifold which we call the collision-ejection manifold. (2) We use the collision-ejection manifold to show geometrically, without writing down any power series, that SBC solutions can be collectively analytically continued. That is, all SBC orbits, not just a single orbit, can be written as a convergent power series in s=t1∕3 with coefficients that depend real analytically on initial conditions that lie in a real analytic submanifold. (3) There are two important ingredients in our work. (i) We use the intrinsic energies and properly rescaled intrinsic angular momenta of the binaries as variables in order to reduce the order of the singularity and to parametrize (distinguish between different) collision orbits that constitute the stable manifolds of the rest points that appear on the collision manifold in the McGehee coordinates. (ii) We use what we call the Kustaanheimo–Stiefel-projective transformation near a SBC singularity to resolve the singularity and isolate collision and ejection orbits from nearby near-collision and near-ejection orbits. We will see that quaternionic multiplication and quaternionic projective spaces are not suitable.

REAL n-DIMENSIONAL QR-SUBMANIFOLDS OF MAXIMAL QR-DIMENSION IMMERSED IN QP(n+p)/4

The purpose of this paper is to study n-dimensional QR-submanifolds of (p-1) QR-dimension immersed in a quaternionic projective space <TEX>$QP^{(n+p)/4}$</TEX> of constant Q-sectional curvature 4 and especially to determine such submanifolds under the additional condition concerning with shape operator.

E7, Wirtinger inequalities, Cayley 4-form, and homotopy

We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalization of the Wirtinger inequality for the comass. Using a model for the classifying space BS3 built inductively out of BS1, we prove that the symmetric metrics of certain two-point homogeneous manifolds turn out not to be the systolically optimal metrics on those manifolds. We point out the unexpected role played by the exceptional Lie algebra E7 in systolic geometry, via the calculation of Wirtinger constants. Using a technique of pullback with controlled systolic ratio, we calculate the optimal systolic ratio of the quaternionic projective plane, modulo the existence of a Joyce manifold with Spin(7)-holonomy and unit middle-dimensional Betti number

Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds

We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Weinberger and to Yang, but which depend in an explicit way on the mean curvature. In later sections, we prove similar results for Schrödinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as for the Kohn Laplacian on subdomains of the Heisenberg group. Among the consequences of this analysis are an extension of Reilly’s inequality, bounding any eigenvalue of the Laplacian in terms of the mean curvature, and spectral criteria for the immersibility of manifolds in homogeneous spaces.

On harmonic tori in compact rank one symmetric spaces

In this paper we prove that, in contrast with the S n and C P n cases, there are harmonic 2-tori into the quaternionic projective space H P n which are neither of finite type nor of finite uniton number; we also prove that any harmonic 2-torus in a compact Riemannian symmetric space which can be obtained via the twistor construction is of finite type if and only it is constant; in particular, we conclude that any harmonic 2-torus in C P n or S n which is simultaneously of finite type and of finite uniton number must be constant.

Formula omitted]-actions fixing [formula omitted

The main result of this paper is the determination, up to equivariant cobordism, of all manifolds with Z 2 k -action whose fixed point set is F = K d P 2 s ∪ K d P n , where n ⩾ 2 s + 1 is even and s ⩾ 1 . Here, K d P n is the real ( d = 1 ), complex ( d = 2 ) or quaternionic ( d = 4 ) n-dimensional projective space, with real dimension dn. This extends a previous and recent result of the authors, concerning the case d = 1 and s = 1 . We also obtain this equivariant cobordism classification for d = 2 and 4 in the cases F = { point } ∪ K d P n , where n ⩾ 2 is even, and F = K d P m ∪ K d P n , where m is odd and n ⩾ 0 is even; for d = 1 and k = 1 , these results are due to D.C. Royster.

Parabolic Geometries Related with Several Fueter Operators

The theory developed for certain geometric structures (the so-called parabolic geometries) was successfully applied in a study of functions of several quaternionic variables satisfying the Fueter (Dirac) equation in each variable separately. The special case of a parabolic geometry used in this application is the quaternionic structure on a manifold with dimension 4n. A homogeneous model for quaternionic geometry is the quaternionic projective space \({\mathbb{P}}^n({\mathbb{H}})\). A question discussed in the paper is whether similar applications are possible for dimensions different from four.