Quaternionic Manifolds Research Articles

The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven

On a compact quaternionic contact (qc) manifold of dimension bigger than seven and satisfying a Lichnerowicz type lower bound estimate we show that if the rst positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of the qc structure, the manifold is qc equivalent to the standard 3-Sasakian sphere. The same conclusion is shown to hold on a non-compact qc manifold which is complete with respect to the associated Riemannian metric assuming the existence of a function with traceless horizontal Hessian.

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.

On the equivalence of quaternionic contact structures

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.

Partial breaking in the rigid limit of $\mathcal{N}=2$ gauged supergravity

Using a new manner to rescale fields in $\mathcal{N}=2$ gauged supergravity with n$_{V}$ vector multiplets and n$_{H}$ hypermultiplets, we develop the explicit derivation of the rigid limit of quaternionic isometry Ward identities agreeing with known results. We show that the rigid limit can be achieved, amongst others, by performing two successive transformations on the covariantly holomorphic sections $V^{M}\left( z,\bar{z}\right) $ of the special Kahler manifold: a particular symplectic change followed by a particular Kahler transformation. We also give a geometric interpretation of the $\eta_{i}$ parameters used in arXiv:1508.01474 to deal with the expansion of the holomorphic prepotential $\mathcal{F}\left( z\right) $ of the $\mathcal{N}=2$ theory. We give as well a D- brane realisation of gauged quaternionic isometries and an interpretation of the embedding tensor $\vartheta_{M}^{u}$ in terms of type IIA/IIB mirror symmetry. Moreover, we construct explicit metrics for a new family of $4r$- dimensional quaternionic manifolds $\boldsymbol{M}_{QK}^{\left( n_{H}\right) }$ classified by ADE Lie algebras generalising the $SO\left( 1,4\right) /SO\left( 4\right) $ geometry which corresponds to $A_{1}\sim su\left( 2\right) $. The conditions of the partial breaking of $\mathcal{N}=2$ supersymmetry in the rigid limit are also derived for both the observable and the hidden sectors. Other features are also studied.

Convexity of the entropy of positive solutions to the heat equation on quaternionic contact and CR manifolds

A proof of the monotonicity of an entropy like energy for the heat equation on a quaternionic contact and CR manifolds is proven

English

As a generalization of anti-invariant Riemannian submersions and Lagrangian Riemannian submersions, we introduce the notions of h-anti-invariant submersions and h-Lagrangian submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We obtain characterizations and investigate some properties: the integrability of distributions, the geometry of foliations, and the harmonicity of such maps. We also find a condition for such maps to be totally geodesic and give some examples of such maps. Finally, we obtain some types of decomposition theorems.

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.

Quaternionic-like manifolds and homogeneous twistor spaces.

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 manifolds endowed with embedded spheres, thus, emphasizing the abundance of the quaternionic-like manifolds.

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.

A direct approach to quaternionic manifolds

The recent definition of slice regular function of several quaternionic variables suggests a new notion of quaternionic manifold. We give the definition of quaternionic regular manifold, as a space locally modeled on , in a slice regular sense. We exhibit some significant classes of examples, including manifolds which carry a quaternionic affine structure.

Anti-invariant Riemannian Submersions: A Lie-theoretical Approach

We give a construction which is Lie theoretic of anti-invariant Riemannian submersions from almost Hermitian manifolds, from quaternion manifolds, from para-Hermitian manifolds, from para-quaternion manifolds, and from octonian manifolds. This yields many compact Einstein examples.

Balls in quaternionic hyperbolic manifolds

By use of the Zassenhaus neighborhood of Sp(n,1), we obtain an explicit lower bound for the radius of the largest inscribed ball in quaternionic hyperbolic n-manifold $\mathscr{M} = \mathbf H_\mathbf H ^n/Γ$. As an application, we obtain a lower bound for the volumes of quaternionic hyperbolic n-manifolds.

Symplectically invariant flow equations for N = 2, D = 4 gauged supergravity with hypermultiplets

We consider N=2 supergravity in four dimensions, coupled to an arbitrary number of vector- and hypermultiplets, where abelian isometries of the quaternionic hyperscalar target manifold are gauged. Using a static and spherically or hyperbolically symmetric ansatz for the fields, a one-dimensional effective action is derived whose variation yields all the equations of motion. By imposing a sort of Dirac charge quantization condition, one can express the complete scalar potential in terms of a superpotential and write the action as a sum of squares. This leads to first-order flow equations, that imply the second-order equations of motion. The first-order flow turns out to be driven by Hamilton's characteristic function in the Hamilton-Jacobi formalism, and contains among other contributions the superpotential of the scalars. We then include also magnetic gaugings and generalize the flow equations to a symplectically covariant form. Moreover, by rotating the charges in an appropriate way, an alternative set of non-BPS first-order equations is obtained that corresponds to a different squaring of the action. Finally, we use our results to derive the attractor equations for near-horizon geometries of extremal black holes.

Moduli spaces of AdS5 vacua in N $$ \mathcal{N} $$ = 2 supergravity

We determine the conditions for maximally supersymmetric AdS_5 vacua of five-dimensional gauged N=2 supergravity coupled to vector-, tensor- and hypermultiplets charged under an arbitrary gauge group. In particular, we show that the unbroken gauge group of the AdS_5 vacua has to contain an U(1)_R-factor. Moreover we prove that the scalar deformations which preserve all supercharges form a Kahler submanifold of the ambient quaternionic Kahler manifold spanned by the scalars in the hypermultiplets.

Three-dimensional quaternionic condensations, Hopf invariants, and skyrmion lattices with synthetic spin-orbit coupling

We study the topological configurations of the two-component condensates of bosons with the three-dimensional $(3\mathrm{D}) \stackrel{P\vec}{\ensuremath{\sigma}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{p}$ Weyl-type spin-orbit coupling subject to a harmonic trapping potential. The topology of the condensate wave functions manifests in the quaternionic representation. In comparison to the $\text{U}(1)$ complex phase, the quaternionic phase manifold is ${S}^{3}$ and the spin orientations form the ${S}^{2}$ Bloch sphere through the first Hopf mapping. The spatial distributions of the quaternionic phases exhibit the 3D skyrmion configurations, and the spin distributions possess nontrivial Hopf invariants. Spin textures evolve from the concentric distributions at the weak spin-orbit coupling regime to the rotation symmetry-breaking patterns at the intermediate spin-orbit coupling regime. In the strong spin-orbit coupling regime, the single-particle spectra exhibit the Landau-level-type quantization. In this regime, the three-dimensional skyrmion lattice structures are formed when interactions are below the energy scale of Landau-level mixings. Sufficiently strong interactions can change condensates into spin-polarized plane-wave states or superpositions of two plane waves exhibiting helical spin spirals.

Stochastic growth dynamics and composite defects in quenched immiscible binary condensates

We study the topological configurations of the two-component condensates of bosons with the $3$D $\vec{\sigma}\cdot \vec{p}$ Weyl-type spin-orbit coupling subject to a harmonic trapping potential. The topology of the condensate wavefunctions manifests in the quaternionic representation. In comparison to the $U(1)$ complex phase, the quaternionic phase manifold is $S^3$ and the spin orientations form the $S^2$ Bloch sphere through the 1st Hopf mapping. The spatial distributions of the quaternionic phases exhibit the 3D skyrmion configurations, and the spin distributions possess non-trivial Hopf invariants. Spin textures evolve from the concentric distributions at the weak spin-orbit coupling regime to the rotation symmetry breaking patterns at the intermediate spin-orbit coupling regime. In the strong spin-orbit coupling regime, the single-particle spectra exhibit the Landau-level type quantization. In this regime, the three-dimensional skyrmion lattice structures are formed when interactions are below the energy scale of Landau level mixings. Sufficiently strong interactions can change condensates into spin-polarized plane-wave states, or, superpositions of two plane-waves exhibiting helical spin spirals.

Quaternionic contact Einstein manifolds

The main result is that the qc-scalar curvature of a seven dimensional quaternionic contact Einstein manifold is a constant. In addition, we characterize qc-Einstein structures with certain flat vertical connection and develop their local structure equations. Finally, regular qc-Ricci flat structures are shown to fibre over hyper-Kahler manifolds.

On the quaternionic manifolds whose twistor spaces are Fano manifolds

Let $M$ be a quaternionic manifold, $\dim M=4k$, whose twistor space is a Fano manifold. We prove the following: (a) $M$ admits a reduction to ${\rm Sp}(1)\times{\rm GL}(k,\mathbb{H})$ if and only if $M=\mathbb{H} P^k$, (b) either $b_2(M)=0$ or $M={\rm Gr}_2(k+2,\mathbb{C})$. This generalizes results of S. Salamon and C. R. LeBrun, respectively, who obtained the same conclusions under the assumption that $M$ is a complete quaternionic-Kähler manifold with positive scalar curvature.

Almost h-semi-slant Riemannian maps to almost quaternionic Hermitian manifolds

We introduce the notions of almost h-slant Riemannian maps, almost h-semi-invariant Riemannian maps, and almost h-semi-slant Riemannian maps from Riemannian manifolds to almost quaternionic Hermitian manifolds. We investigate the harmonicity of such maps and the geometry of distributions. We also find the conditions for such maps to be totally geodesic, relate the notion of pseudo-horizontally weakly conformal maps to those notions, and give some examples of such maps.

The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds

We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz–Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the Riemannian case highlights new and old ideas which are then developed in the settings of Iwasawa sub-Riemannian geometries.