Curvature vs. Almost Hermitian Structures

A Bochner-type formula for almost Hermitian manifolds is introduced. From this formula, one can find obstructions imposed by the curvature to the existence of certain almost Hermitian structures on compact manifolds.     

The Kloosterman problem for binary Hermitian lattices

A Hermitian lattice over an imaginary quadratic field $\mathbb {Q}(\sqrt{-m})$ is called almost universal if it represents all but finitely many positive integers. We investigate almost universal binary Hermitian lattices and provide a Bochnak-Oh type criterion on almost universality. In particular, all almost universal $p$ -anisotropic binary Hermitian lattices are universal, and we give the complete list of all such Hermitian lattices.     

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.     

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 the Twistor Spaces of Almost Hermitian Manifolds

We study Grassmannian bundles G_k(M) of analytical 2k-planes over an almost Hermitian manifold M²ⁿ, from the point of view of the generalized twistor spaces of [13], and with the method of the moving frame [9]. G₁(M⁴) is the classical twistor space. We find four distinguished almost Hermitian structures, one of them being that of [13], and discuss their integrability and Kählerianity. For n=2, we compute the corresponding Hermitian connections, and derive consequences about the corresponding first Chern classes.     

Hessian Comparison and Spectrum Lower Bound of Almost Hermitian Manifolds

The authors obtain a complex Hessian comparison for almost Hermitian manifolds, which generalizes the Laplacian comparison for almost Hermitian manifolds by Tossati, and a sharp spectrum lower bound for compact quasi Kähler manifolds and a sharp complex Hessian comparison on nearly Kähler manifolds that generalize previous results of Aubin, Li Wang and Tam-Yu.     

On conformally flat almost Hermitian manifolds

We study some of 2n-dimensional conformally flat almost Hermitian manifolds with J-(anti)-invariant Ricci tensor. Received 13 May 2000; revised 15 February 2001.     

关于近Hermitian流形的注记(英文)

We give a necessary and sufficient condition for an almost Hermitian manifold to be a Kahler manifold. By making use of this condition, we give a new proof of Goldberg＇s theorem.     

The Yamabe problem for almost Hermitian manifolds

The conformal class of a Hermitian metric g on a compact almost complex manifold (M^2m, J) consists entirely of metrics that are Hermitian with respect to J. For each one of these metrics, we may define a J-twisted version of the Ricci curvature, the J-Ricci curvature, and its corresponding trace, the J-scalar curvature s^J. We ask if the conformal class of g carries a metric with constant s^J, an almost Hermitian version of the usual Yamabe problem posed for the scalar curvature s. We answer our question in the affirmative. In fact, we show that (2m−1)s^J−s=2(2m−1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g. Using techniques developed for the solution of the problem for s, we construct an almost Hermitian Yamabe functional and its corresponding conformal invariant. This invariant is bounded from above by a constant that only depends on the dimension of M, and when it is strictly less than the universal bound, the problem has a solution that minimizes the almost complex Yamabe functional. By the relation above, we see that when W (ω, ω) is negative at least one point, or identically zero, our problem has a solution that minimizes the almost Hermitian Yamabe functional, and the universal bound is reached only in the case of the standard 6-sphere equipped with a suitable almost complex structure. When W(ω, ω) is non-negative and not identically zero, we prove that the conformal invariant is strictly less than the universal bound, thus solving the problem for this type of manifolds as well. We discuss some applications.     

Generalized Almost Hermitian Spaces and Holomorphically Projective Mappings

Mediterranean Journal of Mathematics - We derive some interesting properties of generalized almost Hermitian spaces. Additionally, we study basic equations of holomorphically projective mappings...     

Christoffel Transforms and Hermitian Linear Functionals

In this contribution we are focused on some spectral transformations of Hermitian linear functionals. They are the analogues of the Christoffel transform for linear functionals, i. e. for Jacobi matrices which has been deeply studied in the past. We consider Hermitian linear functionals associated with a probability measure supported on the unit circle. In such a case we compare the Hessenberg matrices associated with such a probability measure and its Christoffel transform. In this way, almost unitary matrices appear. We obtain the deviation to the unit matrix both for principal submatrices and the complete matrices respectively.     

On the Constant Type of Almost Hermitian Manifolds

We suggest a most natural generalization of the notion of constant type for nearly Kählerian manifolds introduced by A. Gray to arbitrary almost Hermitian manifolds. We prove that the class of almost Hermitian manifolds of zero constant type coincides with the class of Hermitian manifolds. We show that the class of G ₁-manifolds of zero constant type coincides with the class of 6-dimensional G ₁-manifolds with a non-integrable structure. Finally, we prove that the class of normal G ₂-manifolds of nonzero constant type coincides with the class of 4-dimensional G ₂-manifolds with a nonintegrable structure.     

On the Kenmotsu Hypersurfaces of Special Hermitian Manifolds

We consider almost contact metric hypersurfaces of almost Hermitian manifolds of class W₃ (in the Gray–Hervella terminology). We establish a criterion for minimality of such hypersurfaces in the case when the contact metric structure is cosymplectic.     

On Hermitian curvature flow on almost complex manifolds

In the present paper we generalize the Hermitian curvature flow introduced and studied in Streets and Tian (2011) [6] to the almost complex case.     

Modified Hermitian and skew‐Hermitian splitting methods for non‐Hermitian positive‐definite linear systems

To further study the Hermitian and non‐Hermitian splitting methods for a non‐Hermitian and positive‐definite matrix, we introduce a so‐called lopsided Hermitian and skew‐Hermitian splitting and then establish a class of lopsided Hermitian/skew‐Hermitian (LHSS) methods to solve the non‐Hermitian and positive‐definite systems of linear equations. These methods include a two‐step LHSS iteration and its inexact version, the inexact Hermitian/skew‐Hermitian (ILHSS) iteration, which employs some Krylov subspace methods as its inner process. We theoretically prove that the LHSS method converges to the unique solution of the linear system for a loose restriction on the parameter α. Moreover, the contraction factor of the LHSS iteration is derived. The presented numerical examples illustrate the effectiveness of both LHSS and ILHSS iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

On balanced Hermitian structures on Lie groups

We present several methods for the construction of balanced Hermitian structures on Lie groups. In our methods a partial differential equation is involved so that the resulting structures are in general non homogeneous. In particular, we prove that for 3-step nilpotent Lie groups G of dimension 6, any left-invariant complex structure on G admits a balanced Hermitian metric. Starting from normal almost contact structures, we construct balanced metrics on 6-dimensional manifolds, generalizing warped products. Finally, explicit balanced Hermitian structures are also given on solvable Lie groups defined as semidirect products ${\mathbb{R}^k \ltimes \mathbb{R}^{2n-k}}$ .     

On Almost Hermitian 4-manifolds with J-invariant Ricci Tensor

We prove that every almost Hermitian 4-manifold with J-invariant Ricci tensor which is conformally flat or has harmonic curvature is either a space of constant curvature or a Kähler manifold. We also obtain analogous results on almost Kähler 4-manifolds.     

Chern-flat and Ricci-flat invariant almost Hermitian structures

We study left-invariant almost Hermitian structures on homogeneous spaces having either flat Chern connection or flat Ricci–Chern form. Many examples are carefully described, and a classification is given in low dimensions.     

The sixteen classes of almost Hermitian manifolds and their linear invariants

Summary It is shown that in a natural way there are precisely sixteen classes of almost Hermitian manifolds.     

Dirichlet problem for Hermitian-Einstein equation over almost Hermitian manifold

In this paper, we investigate the Dirichlet problem for Hermitian-Einstein equation on complex vector bundle over almost Hermitian manifold, and we obtain the unique solution of the Dirichlet problem for Hermitian-Einstein equation.