A quaternionic Kähler manifold M is called {\it positive} if it has positive scalar curvature. The main purpose of this paper is to prove several connectedness theorems for quaternionic immersions in a quaternionic Kähler manifold, e.g. the Barth-Lefschetz type connectedness theorem for quaternionic submanifolds in a positive quaternionic Kähler manifold. As applications we prove that, among others, a 4m-dimensional positive quaternionic Kähler manifold with symmetry rank at least (m-2) must be either isometric to \Bbb HP^m or Gr_2(\Bbb C^{m+2}), if m\ge 10.
Let M be a positive quaternionic Kaehler manifold of dimension 4m. If the isometry group Isom(M) has rank at least m/2 +3, then M is isometric to HP^m or Gr_2(C^{m+2}). The lower bound for the rank is optimal if m is even.
We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to the quaternionic projective space or to the quaternionic hyperbolic space. We describe locally the hyper-Hermitian quaternionic Kaehler manifolds with closed Lee form and show that the only complete simply connected such manifold is the quaternionic hyperbolic space.
In this paper we study 3-submersions from a QR-hypersurface of a quaternionic Kaehler manifold onto an almost quaternionic hermitian manifold. We also prove the non-existence of quaternionic submersions between quaternionic Kaehler manifolds which are not locally hyper-Kaehler.
Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kaehler manifolds are considered. Some necessary and sufficient conditions the investigated manifolds be isotropic hyper-Kaehlerian and flat are found. It is proved that the quaternionic Kaehler manifolds with the considered metric structure are Einstein for dimension at least 8. The class of the non-hyper-Kaehler quaternionic Kaehler manifold of the considered type is determined.
An explicit classification of homogeneous quaternionic Kaehler structures by real tensors is derived and we relate this to the representation-theoretic description found by Fino. We then show how the quaternionic hyperbolic space HH(n) is characterised by admitting homogeneous structures of a particularly simple type. In the process we study the properties of different homogeneous models for HH(n).
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 construct left invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven dimensional manifold, equipped with a certain qc structure, has a quaternionic Kaehler metric as well as a metric with holonomy contained in Spin(7). As a consequence we determine explicit quaternionic Kaehler metrics and Spin(7)-holonomy metrics which seem to be new. Moreover, we give expl…
In this paper, authors have established Chen's inequalities for the submanifolds of quaternionic Kaehler manifolds characterized by Ricci quarter-symmetric metric connection. Other than these inequalities, we have also derived generalized normalized Casorati curvature inequalities for the same submanifolds.
We consider the Dirac operator on compact quaternionic Kaehler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space.
We discuss a peculiar interplay between the representation theory of the holonomy group of a Riemannian manifold, the Weitzenboeck formula for the Hodge-Laplace operator on forms and the Lichnerowicz formula for twisted Dirac operators. For quaternionic Kaehler manifolds this leads to simple proofs of eigenvalue estimates for Dirac and Laplace operators. Moreover it enables us to determine which representations can contribute to harmonic forms. As a corollary we prove the vanishing of certain odd Betti numbers on compact quaternionic Kaehler manifolds of negative scalar curvature. We simplify …
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…
We classify isometric actions of compact Lie groups on quaternionic-Kähler projective spaces with vanishing homogeneity rank. We also show that they are not in general quaternion-coisotropic.
In a previous paper we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kaehler manifolds. In the present article we show that the only manifolds in the limit case, i.e. the only manifolds where the lower bound is attained as an eigenvalue, are the quaternionic projective spaces. We use the equivalent formulation in terms of the quaternionic Killing equation and show that a nontrivial solution defines a parallel spinor on the associated hyperkaehler manifold.
BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M x N of a quaternionic-Kaehler manifold M of negative scalar curvature and a very special real manifold N of dimension n >=0. Such gradient flows are generated by the `energy function' f = P^2, where P is a (bundle-valued) moment map associated to n+1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kaehler manifolds. For the homogeneous quaternionic-Kaehler manifolds we prove more specific results depen…
A universal lower bound for the first positive eigenvalue of the Dirac operator on a compact quaternionic Kaehler manifold M of positive scalar curvature is calculated. It is shown that it is equal to the first positive eigenvalue on the quaternionic projective space. For this, the horizontal tangent bundle on the canonical SO(3)-bundle over M is equipped with a hyperkaehlerian structure and the corresponding splitting of the horizontal spinor bundle is considered. The desired estimate is obtained by looking at hyperkaehlerian twistor operators on horizontal spinors.
We study a class of supersymmetric spinning particle models derived from the radial quantization of stationary, spherically symmetric black holes of four dimensional N= 2 supergravities. By virtue of the c-map, these spinning particles move in quaternionic Kaehler manifolds. Their spinning degrees of freedom describe mini-superspace-reduced supergravity fermions. We quantize these models using BRST detour complex technology. The construction of a nilpotent BRST charge is achieved by using local (worldline) supersymmetry ghosts to generating special holonomy transformations. (An interesting byp…
In a previous article we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kaehler manifolds. In the present article we study the limiting case, i. e. manifolds where the lower bound is attained as an eigenvalue. We give an equivalent formulation in terms of a quaternionic Killing equation and show that the only symmetric quaternionic Kaehler manifolds with smallest possible eigenvalue are the quaternionic projective spaces.
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…
Positive Quaternion Kaehler Manifolds are Riemannian manifolds with holonomy contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they are symmetric spaces. We prove this conjecture in dimension 20 under additional assumptions and we provide recognition theorems for quaternionic projective spaces (in low dimensions) as well as the real Grassmanian (which is Positive Quaternion Kaehler).
In a general and non metrical framework, we introduce the class of co-CR quaternionic manifolds, which contains the class of quaternionic manifolds, whilst in dimension three it particularizes to give the Einstein-Weyl spaces. We show that these manifolds have a rich natural Twistor Theory and, along the way, we obtain a heaven space construction for quaternionic-Kaehler manifolds.
We review the map between hypercomplex manifolds that admit a closed homothetic Killing vector (i.e. `conformal hypercomplex' manifolds) and quaternionic manifolds of 1 dimension less. This map is related to a method for constructing supergravity theories using superconformal techniques. An explicit relation between the structure of these manifolds is presented, including curvatures and symmetries. An important role is played by `ξtransformations', relating connections on quaternionic manifolds, and a new type `\hatξtransformations' relating complex structures on conformal hypercomplex manifol…
We consider strict and complete nearly Kaehler manifolds with the canonical Hermitian connection. The holonomy representation of the canonical Hermitian connection is studied. We show that a strict and complete nearly Kaehler is locally a Riemannian product of homogenous nearly Kaehler spaces, twistor spaces over quaternionic Kaehler manifolds and 6-dimensional nearly Kaehler manifolds. As an application we obtain structure results for totally geodesic Riemannian foliations admitting a compatible Kaehler structure. Finally, we obtain a classification result for the homogenous case, reducing a …
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.
In this article we prove an upper bound for a Hilbert polynomial on quaternionic Kaehler manifolds of positive scalar curvature. As corollaries we obtain bounds on the quaternionic volume and the degree of the associated twistor space. Moreover the article contains some details on differential equations of finite type. Part of it is used in the proof of the main theorem.
We consider complete nearly Kähler manifolds with the canonical Hermitian connection. We prove some metric properties of strict nearly Kähler manifolds and give a sufficient condition for the reducibility of the canonical Hermitian connection. A holonomic condition for a nearly Kähler manifold to be a twistor space over a quaternionic Kähler manifold is given. This enables us to give classification results in 10-dimensions.
A hyperkaehler manifold with a circle action fixing just one complex structure admits a natural a hyperholomorphic line bundle. This forms the basis for the construction of a corresponding quaternionic Kaehler manifold in the work of A.Haydys. We construct in this paper the corresponding holomorphic line bundle on twistor space and compute many examples, including monopole and Higgs bundle moduli spaces. We also show that the bundle on twistor space has a natural meromorphic connection which realizes it as the quantum line bundle for the hyperkaehler family of holomorphic symplectic structures…
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 $Sp(1) \times GL(k,H)$ if and only if $M=HP^k$, (b) either $b_2(M)=0$ or $M=Gr_2(k+2,C)$. 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-Kaehler manifold with positive scalar curvature.
We develop and study quaternionic and octonionic analogies of Cartan angular and Toledo invariants that are well known in the complex hyperbolic space. Using such invariants we study quasifuchsian deformations (including bendings) of quaternionic and octonionic hyperbolic manifolds.
We construct examples of $S^1$-manifolds with finite second homotopy group and non-vanishing $\hat A$-genus. This is related to the classification of positive quaternionic Kaehler manifolds.
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…
We show that, in quaternionic geometry, the Ward transform is a manifestation of the functoriality of the basic correspondence between the $ρ$-quaternionic manifolds and their twistor spaces. We apply this fact, together with the Penrose transform, to obtain existence results for hypercomplex manifolds and for harmonic morphisms from hyper-Kaehler manifolds.
We consider locally conformal Kaehler geometry as an equivariant, homothetic Kaehler geometry (K,Γ). We show that the de Rham class of the Lee form can be naturally identified with the homomorphism projecting Γto its dilation factors, thus completing the description of locally conformal Kaehler geometry in this equivariant setting. The rank r of a locally conformal Kaehler manifold is the rank of the image of this homomorphism. Using algebraic number theory, we show that r is non-trivial, providing explicit examples of locally conformal Kaehler manifolds with 1<r<b_1. As far as we know, these …
We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces of dimension greater than two. For the quaternionic hyperbolic spaces of dimension greater than two we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Elie Cartan.
We classify all holomorphic actions of higher rank lattices on compact Kaehler manifolds of dimension 3. This provides a complete answer to Zimmer's program for holomorphic actions on compact Kaehler manifolds of dimension at most 3.
We show that generic rank conditions on the second fundamental form of an isometric immersion $f\colon M^{2n}\to\mathbb{R}^{2n+p}$ of a Kaehler manifold of complex dimension $n\geq 2$ into Euclidean space with low codimension $p$ implies that the submanifold has to be minimal. If $M^{2n}$ if simply connected, this amounts to the existence of a one-parameter associated family of isometric minimal immersions unless $f$ is holomorphic.
We introduce the notion of CR quaternionic map and we prove that any such real-analytic map, between CR quaternionic manifolds, is the restriction of a quaternionic map between quaternionic manifolds. As an application, we prove, for example, that for any submanifold $M$, of dimension $4k-1$, of a quaternionic manifold $N$, such that $TM$ generates a quaternionic subbundle of $TN|_M$, of (real) rank $4k$, there exists, locally, a quaternionic submanifold of $N$, containing $M$ as a hypersurface.
We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one: 1) A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic. 2) 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 the quaternionic maps between open sets of quaternionic projective spaces.
We conclude the classification of cohomogeneity one actions on symmetric spaces of rank one by classifying cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. As a by-product of our proof, we produce uncountably many examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces.
3-quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. This paper throws new light on their geometric structure which reveals to be generally richer compared to the 3-Sasakian subclass. In fact, it turns out that they are multiply foliated by four distinct fundamental foliations. The study of the transversal geometries with respect to these foliations allows us to link the 3-quasi-Sasakian manifolds to the more famous hyper-Kaehler and quaternionic-Kaehler geometries. Furthermore, we stron…
We study the problem of existence of Kähler--Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree $22$ that admit a faithful action of the multiplicative group $\mathbb{C}^\ast$. We prove that, except possibly two explicitly described cases, all such smooth Fano threefolds are Kähler--Einstein.
We define the C^*-action on moduli spaces of reductive representations of fundamental groups of quasi-compact Kaehler manifolds by solving Hermitian-Yang-Mills equation. As applications in algebraic geometry we show a non-abelian Hodge (p,q)-type theorem for families of quasi-projective manifolds. We also prove that any discrete sub group of a Lie group of rank >1 that is not of Hodge type can't be the fundamental group of a quasi-compact Kaehler manifold.
We classify those manifolds mentioned in the title which have finite topological type. Namely we show any such connected M is isomorphic to a hyperkaehler quotient of a flat quaternionic vector space by an abelian group. We also show that a compact connected and simply connected 3-Sasakian manifold of dimension 4n-1 whose isometry group has rank n+1 is isometric to a 3-Sasakian quotient of a sphere by a torus. As a corollary, a compact connected quaternion-Kaehler 4n-manifold with positive scalar curvature and isometry group of rank n+1 is isometric to the quaternionic projective space or the …
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…
We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m>4) of positive sectional curvature on which a (m-1)-torus acts isometrically is homeomorphic to a complex projective space if and only if its Euler characteristic is not 2. By a result of Wilking, these results imply a homeomorphism classification for positively curved n-manifolds (n>7) of almost maxi…
The geometry arising from Michelson & Strominger's study of N=4B supersymmetric quantum mechanics with superconformal D(2,1;alpha)-symmetry is a hyperKaehler manifold with torsion (HKT) together with a special homothety. It is shown that different parameters alpha are related via changes in potentials for the HKT target spaces. For alpha not 0 or -1, we describe how each such HKT 4m-manifold M is derived from a space N of dimension 4m-4 which is quaternionic Kaehler with torsion and carries an Abelian instanton.
We prove that any compact selfdual Einstein 4-orbifold of positive scalar curvature whose isometry group contains a 2-torus is, up to an orbifold covering, a quaternion Kaehler quotient of (k-1)-dimensional quaternionic projective space by a (k-2)-torus for some $k\geq 2$. We also obtain a topological classification in terms of the intersection form of the 4-orbifold.
Austere submanifolds of Euclidean space were introduced in 1982 by Harvey and Lawson in their foundational work on calibrated geometries. In general, the austerity condition is much stronger than minimality since it express that the nonzero eigenvalues of the shape operator of the submanifold appear in opposite pairs for any normal vector at any point. Thereafter, the challenging task of finding non-trivial explicit examples, other than minimal immersions of Kaehler manifolds, only turned out submanifolds of rank two, and these are of limited interest in the sense that in this special situatio…
Let $M$ be a complete Riemannian manifold and suppose $p\in M$. For each unit vector $v \in T_p M$, the $\textit{Jacobi operator}$, $\mathcal{J}_v: v^\perp \rightarrow v^\perp$ is the symmetric endomorphism, $\mathcal{J}_v(w) = R(w,v)v$. Then $p$ is an $\textit{isotropic point}$ if there exists a constant $κ_p \in \mathbf{R}$ such that $\mathcal{J}_v = κ_p \textit{ Id}_{v^\perp}$ for each unit vector $v \in T_pM$. If all points are isotropic, then $M$ is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. …
The aim of this article is the proof of the following result: Let M be a connected manifold endowed with a regular Cartan geometry modelled on the boundary X of the d-dimensional real (resp. complex, resp. quaternionic, resp. octonionic) hyperbolic space. If the group of automorphisms of M does not act properly on M, then M is geometrically isomorphic to: - X if M is compact. - X minus a point in the other cases.
We construct infinite families of regular normal Cartan geometries with nonvanishing curvature and essential automorphisms on closed manifolds for many higher rank parabolic model geometries. To do this, we use particular elements of the kernel of the Kostant Laplacian to construct homogeneous Cartan geometries of the desired type, giving a global realization of an elegant local construction due to Kruglikov and The, and then modify these homogeneous geometries to make their base manifolds compact. As a demonstration, we apply the construction to quaternionic contact structrures of mixed signa…
Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=π. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and positive spherical rank, then M is isometric to a compact, rank one symmetric space (CROSS) i.e., isometric to a sphere, complex projective space, quaternionic projective space or to the Cayley plane. The notion of spherical rank is analogous to the notions of Euclidean rank and hy…
