Curvature identities on almost Hermitian manifolds and applications

We systematically derive the Bianchi identities for the canonical connection on an almost Hermitian manifold.Moreover,we also compute the curvature tensor of the Levi-Civita connection on almost Hermitian manifolds in terms of curvature and torsion of the canonical connection.As applications of the curvature identities,we obtain some results about the integrability of quasi K¨ahler manifolds and nearly K¨ahler manifolds.     

On almost hyperHermitian structures on Riemannian manifolds and tangent bundles

Some results concerning almost hyperHermitian structures are considered, using the notions of the canonical connection and the second fundamental tensor field h of a structure on a Riemannian manifold which were introduced by the second author. With the help of any metric connection on an almost Hermitian manifold M an almost hyperHermitian structure can be constructed in the defined way on the tangent bundle TM. A similar construction was considered in [6], [7]. This structure includes two basic anticommutative almost Hermitian structures for which the second fundamental tensor fields h ¹ and h ² are computed. It allows us to consider various classes of almost hyperHermitian structures on TM. In particular, there exists an infinite-dimensional set of almost hyperHermitian structures on TTM where M is any Riemannian manifold.     

Affine connections on almost para-cosymplectic manifolds

Identities for the curvature tensor of the Levi-Cività connection on an almost para-cosymplectic manifold are proved. Elements of harmonic theory for almost product structures are given and a Bochner-type formula for the leaves of the canonical foliation is established.     

Sur la géométrie systolique des variétés de Bieberbach

We review the theory of quaternionic Kähler and hyperkähler structures. Then we consider the tangent bundle of a Riemannian manifold M endowed with a metric connection D, with torsion, and with its well estabilished canonical complex structure. With an almost Hermitian structure on M it is possible to find a quaternionic Hermitian structure on TM, which is quaternionic Kähler if, and only if, D is flat and torsion free. We also review the symplectic nature of TM, in the wider context of geometry with torsion. Finally we discover an S ³-bundle of complex structures, which expands to TM the well known S ²-twistor bundle of a quaternionic Hermitian manifold M.     

Curvature identities for principle <Emphasis Type="Italic">T</Emphasis><Superscript>1</Superscript>-bundles over almost Hermitian manifolds

We show that the identities R ₁, R ₂ and R ₃ for an almost Hermitian structure S on the base of the canonical principal T ¹-bundle are equivalent to their contact analogs for the induced almost contact metric structure S ^# on the total space of this bundle. We prove that the canonical connection of the canonical principal T ¹-bundle over a Hermitian or a quasi-Kähler manifold of class R ₃ is normal. We also prove that that the canonical connection of the canonical principal T ¹-bundle over a Vaisman-Gray manifold M of class R ₃ is normal if and only if the Lee vector of M belongs to the center of the adjoint K-algebra.     

On Nearly-Kähler Geometry

We consider complete nearly-Kähler manifolds with a canonicalHermitian 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.     

Local Riemann-Roch theorem for almost Hermitian manifolds

In this paper, we give a local Riemann-Roch theorem for certain almost Hermitian manifolds by a canonical connection and the theory of almost Hermitian structure of null eccentricity. In particular, we obtain a local Riemann-Roch theorem for all closed almost complex manifolds of dimension 4. Also, by using the heat equation method, we give an analytic approach to a local index theorem for generalized Dirac operators.     

Twistorial examples of almost Hermitian manifolds with Hermitian Ricci tensor

We construct new twistorial examples of non-Kähler almost Hermitian manifolds with Hermitian Ricci tensor by means of a natural almost Hermitian structures on the twistor space of an almost Hermitian four manifold.     

On Paraquaternionic Submersions Between Paraquaternionic Kähler Manifolds

In this paper we deal with some properties of a class of semi-Riemannian submersions between manifolds endowed with paraquaternionic structures, proving a result of non-existence of paraquaternionic submersions between paraquaternionic Kähler non-locally hyper para-Kähler manifolds. Then we examine, as an example, the canonical projection of the tangent bundle, endowed with the Sasaki metric, of an almost paraquaternionic Hermitian manifold.     

Abelian Hermitian geometry

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a Euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.     

Effective estimates on the very ampleness of the canonical line bundle of locally Hermitian symmetric spaces

We study the problem about the very ampleness of the canonical line bundle of compact locally Hermitian symmetric manifolds of non-compact type. In particular, we show that any sufficiently large unramified covering of such manifolds has very ample canonical line bundle, and give estimates on the size of the covering manifold, which is itself a locally Hermitian symmetric manifold, in terms of geometric data such as injectivity radius or degree of coverings.

     

Nearly Kähler 6-Manifolds with Reduced Holonomy

We consider a complete six-dimensional nearly Kählermanifold together with the first canonical Hermitian connection. We showthat if the holonomy of this connection is reducible, then the manifoldendowed with a modified metric and almost complex structure is aKählerian twistor space. This result was conjectured byReyes-Carrión.     

Compact Hyperhermitian–Weyl and Quaternion Hermitian–Weyl Manifolds

Let (M, g, H) be a quaternion Hermitian manifold. The additional datum of a torsion-free connection D preserving both the quaternionic structure H and the conformal class of g defines on M the structure of quaternion Hermitian–Weyl manifold. Under the compactness assumption of both M and the leaves of a canonical foliation, M is here shown to project on a locally 3-Sasakian orbifold P. Then M is proved to admit both a compatible global complex structure and a finite covering M carrying a hyperhermitian–Weyl structure. The uniqueness of the Weyl structure compatible with a given quaternion Hermitian metric and some restrictions on the Betti numbers are also obtained.     

Connections on contact manifolds and contact twistor space

In this paper we generalize the definition of symplectic connection to the contact case. It turns out that any odd-dimensional manifold equipped with a contact form admits contact connections and that any Sasakian structure induces a canonical contact connection. Furthermore (as in the symplectic case), any contact connection induces an almost CR structure on the contact twistor space which is integrable if and only if the curvature of the connection is of Ricci-type.     

Geometric methods for construction of quantum gates

The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. These methods are shown to be very useful for the problem of constructing a universal set of gates for quantum computations: the well-known result that the set of all one-bit gates together with almost any one two-bit gate is universal is considered from the control theory viewpoint. Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold, and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action, and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over the Grassmannian and its relation with the Berry phase is considered and investigated for the trajectories of Hamiltonian dynamical systems. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 44, Quantum Computing, 2007.     

A third order dispersive flow for closed curves into almost Hermitian manifolds

We discuss a short-time existence theorem of solutions to the initial value problem for a third order dispersive flow for closed curves into a compact almost Hermitian manifold. Our equations geometrically generalize a physical model describing the motion of vortex filament. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one derivative arises from the covariant derivative of the almost complex structure. To overcome this difficulty, we introduce a bounded pseudodifferential operator acting on sections of the pullback bundle, and eliminate the loss of one derivative from the partial differential equation of the dispersive flow.     

On π 2 almost geodesic mappings of almost Hermitian manifolds

We consider a class of almost geodesic mappings, namely, almost geodesic mappings of class π ₂, and obtain conditions under which almost Hermitian manifolds admit almost geodesic mappings of class π ₂. We prove that an almost Hermitian manifold admits a π ₂-mapping with respect to a Riemannian connection if and only if it is an NK-manifold. We obtain a condition on the defining form ψ of any nontrivial π ₂(e)-mapping under which a properNK-structure is taken to a proper NK-structure.     

Riemannian foliations on quaternion CR-submanifolds of an almost quaternion Kähler product manifold

The purpose of this paper is to study the canonical foliations of a quaternion CR-submanifold of an almost quaternion Kähler product manifold.     

A new class of almost complex structures

The purpose of this paper is to introduce a new class of almost complex structures J on a Riemannian manifold M by using a certain identity for the relationship between the tensor F _i ^j of J and the Riemann curvature tensor R _hijk of M. This class contains the Kählerian structures, and its relationship with some known classes of almost Hermitian structures defined by similar identities is discussed. For convenience we call each structure of this new class an almost C-structure, and a manifold with an almost C-structure an almost C-manifold. We obtain an analogue of F. Schur's theorem concerning the holomorphic sectional curvature of an almost Hermitian C-manifold, and some sufficient conditions for an almost Hermitian C-manifold to be Kählerian. We show that these results are also true for a manifold with a complex structure.     

Splitting the curvature of the determinant line bundle

It is shown that the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold has a canonical Hermitian metric with compatible connection whose curvature satisfies an additivity formula with contributions from the families of Dirac operators over the two halves.