Chern Connection Research Articles

Quasi-Kähler manifolds with trivial Chern Holonomy

In this paper we study almost complex manifolds admitting a quasi-Kahler Chern-flat metric (Chern-flat means that the holonomy of the Chern connection is trivial). We prove that in the compact case such manifolds are all nilmanifolds. Some partial classification results are established and we prove that a quasi-Kahler Chern-flat structure can be tamed by a symplectic form if and only if the ambient space is isomorphic to a flat torus.

Hermitian curvature flow

We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler–Lagrange equation for this functional is an elliptic equation for Hermitian metrics. Solutions to this equation are related to Kähler–Einstein metrics, and are automatically Kähler–Einstein under certain conditions. Given this, a natural parabolic flow equation arises. We prove short time existence and regularity results for this flow, as well as stability for the flow near Kähler–Einstein metrics with negative or zero first Chern class.

The variation formulas of Finsler submanifolds

Let φ : ( M n , F ) → ( M ¯ n + p , F ¯ ) be an isometric immersion from a Finsler manifold to a Finsler manifold. In this paper, we establish the first variation formula of volume in terms of the Chern connection of M . We also determine the second variation formula of volume for the Landsberg hypersurfaces in Minkowski spaces.

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.

Einstein–Hermitian connection on twisted Higgs bundles

Let X be a smooth projective variety over C . We prove that a twisted Higgs vector bundle ( E , θ ) on X admits an Einstein–Hermitian connection if and only if ( E , θ ) is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved in Wang [10]. Our approach is simpler.

Gray identities, canonical connection and integrability

AbstractWe characterize quasi-Kähler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related to the third Gray identity and in the almost-Kähler case implies the integrability. Our main tool is the existence of generalized holomorphic frames previously introduced by the second author. By using such frames we also give a simpler and shorter proof of a theorem of Goldberg. Furthermore, we study almost-Hermitian structures having the curvature tensor associated to the canonical Hermitian connection equal to zero. We show some explicit examples of quasi-Kähler structures on the Iwasawa manifold having the Hermitian curvature vanishing and the Riemann curvature tensor satisfying the second Gray identity.

On m-Accretivity of Perturbed Bochner Laplacian in L p Spaces on Riemannian Manifolds

We consider a differential expression $${H=\nabla^*\nabla+V}$$ , where $${\nabla}$$ is a Hermitian connection on a Hermitian vector bundle E over a manifold of bounded geometry (M, g) with metric g, and V is a locally integrable section of the bundle of endomorphisms of E. We give a sufficient condition for H to have an m-accretive realization in the space L p (E), where 1 < p < +∞. We study the same problem for the operator Δ M + V in L p (M), where 1 < p < ∞, Δ M is the scalar Laplacian on a complete Riemannian manifold M, and V is a locally integrable function on M.

Critical remarks on Finsler modifications of gravity and cosmology by Zhe Chang and Xin Li

I do not agree with the authors of papers arXiv:0806.2184 and arXiv:0901.1023v1 (published in [Zhe Chang, Xin Li, Phys. Lett. B 668 (2008) 453] and [Zhe Chang, Xin Li, Phys. Lett. B 676 (2009) 173], respectively). They consider that “In Finsler manifold, there exists a unique linear connection – the Chern connection … It is torsion freeness and metric compatibility …”. There are well-known results (for example, presented in monographs by H. Rund and R. Miron and M. Anastasiei) that in Finsler geometry there exist an infinite number of linear connections defined by the same metric structure and that the Chern and Berwald connections are not metric compatible. For instance, the Chern's one (being with zero torsion and “weak” compatibility on the base manifold of tangent bundle) is not generally compatible with the metric structure on total space. This results in a number of additional difficulties and sophistication in definition of Finsler spinors and Dirac operators and in additional problems with further generalizations for quantum gravity and noncommutative/string/brane/gauge theories. I conclude that standard physics theories can be generalized naturally by gravitational and matter field equations for the Cartan and/or any other Finsler metric compatible connections. This allows us to construct more realistic models of Finsler spacetimes, anisotropic field interactions and cosmology.

The geometry of determinant line bundles in noncommutative geometry

This article is concerned with the study of the geometry of determinant line bundles associated to families of spectral triples parametrized by the moduli space of gauge equivalence classes of Hermitian connections on a Hermitian finite projective module. We illustrate our results with some examples that arise in noncommutative geometry.

On the canonical Hermitian connection in nearly Kähler manifolds

In the present paper we prove that the Hermitian curvature tensor $\tilde{R}$ associated to a nearly Kähler metric g always satisfies the second Bianchi identity $\mathfrak{S}(\tilde{\nabla}_X\tilde{R})$ (Y, Z, ·, ·)=0 and that it satisfies the first Bianchi identity $\mathfrak{S}\tilde{R}$(X, Y, Z, ·)=0 if and only if g is a Kähler metric. Furthermore we characterize condition for $\tilde{R}$ to be parallel with respect to the canonical Hermitian connection $\tilde{\nabla}$ in terms of the Riemann curvature tensor and in the last part of the paper we study the curvature of some generalizations of the nearly Kähler structure.

CONCURRENT π-VECTOR FIELDS AND ENERGY β-CHANGE

The present paper deals with an intrinsic investigation of the notion of a concurrent π-vector field on the pullback bundle of a Finsler manifold (M, L). The effect of the existence of a concurrent π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change ([Formula: see text]with[Formula: see text]; [Formula: see text] being a concurrent π-vector field), is established. The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-change of the fundamental linear connection in Finsler geometry: the Cartan connection, the Berwald connection, the Chern connection, and the Hashiguchi connection. Moreover, the change of their curvature tensors is concluded. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

Curvature Properties of the Chern Connection of Twistor Spaces

Curvature Properties of the Chern Connection of Twistor Spaces

Non-Kaehler Heterotic String Compactifications with Non-Zero Fluxes and Constant Dilaton

We construct new explicit compact supersymmetric valid solutions with non-zero field strength, non-flat instanton and constant dilaton to the heterotic equations of motion in dimension six. We present balanced Hermitian structures on compact nilmanifolds in dimension six satisfying the heterotic supersymmetry equations with non-zero flux, non-flat instanton and constant dilaton which obey the three-form Bianchi identity with curvature term taken with respect to either the Levi-Civita, the (+)-connection or the Chern connection. Among them, all our solutions with respect to the (+)-connection on the compact nilmanifold M 3 satisfy the heterotic equations of motion.

A quadratic inequality for sum of co-adjoint orbits

We obtain an e¤ective lower bound on the distance of sum of coadjoint orbits from the origin. Even when the distance is zero, thus the symplectic quotient is well-de ned, our result give a nontrivial constraint on these co-adjoint orbits. In the particular case of unitary groups, we recover the quadratic inequality for eigenvalues of Hermitian matrices satisfying A+B = C: This quadratic inequality was obtained earlier by the authors using completely di¤erent means, namely Klyachko’s theory of toric stable re‡exive sheaves and the Chern number inequality for Hermitian Yang-Mills connection. 1 Introduction Given any rank r Hermitian matrix A, we may order its eigenvalues in such a way that 1 (A) 2 (A) ::: r (A) ; and denote (A) := ( 1 (A) ; ; r (A)) 2 R as its spectrum. In [K2] (Also see [Fu] for an excellent account of the subject), Klyachko discovered following series of linear inequalities for Hermitian matrices A;B;C satisfying A+B = C : X k2K k (C) X i2I i (A) + X j2J j (B) for some triple of subsets I; J;K f1; 2; :::; rg of the same cardinality and such that the associated Schubert cycle sK is a component of sI sJ . This result can be interpreted as describing the linear inequalities which determine the intersection of the sum of co-adjoint orbits O (A) + O (B) + O ( C) with the positive Weyl chamber for the unitary group, which is a convex polytope by Kirwan’s convexity theorem. Klyachko’s result was generalized to any compact Lie group by Berenstein and Sjamaar in the beautiful paper [BeS], they are 1 able to relate above eigenvalue inequalities to the convexity of the image of the moment map. All above inequalities are linear on eigenvalues of matrices. In [LW], we nd a natural quadratic inequality on these eigenvalues by relating the Hermitian matrices to stable re‡exive sheaves over the projective spaces. More precisely, we have shown Theorem 1 Suppose A1; ; AN are rank r Hermitian matrices satisfying N X

A class of isoclinic three-webs

We consider multidimensional isoclinic three-webs with covariantly constant (with respect to the Chern connection) curvature and torsion tensors. It is proved that there exists a unique (up to an isotopy) isoclinic three-web with covariantly constant basic tensors.We find structure and finite equations of this web and consider some its properties.

The flag curvature of invariant (α, β)-metrics of type

In this paper, we study the flag curvature of invariant (α, β)-metrics of the form on homogeneous spaces and Lie groups. We give a formula for the flag curvature of invariant metrics of the form such that α is induced by an invariant Riemannian metric g on the homogeneous space and the Chern connection of F coincides to the Levi-Civita connection of g. Then some conclusions in the cases of naturally reductive homogeneous spaces and Lie groups are given.

Uniform Estimates of Solutions of -Equation on Stein Manifolds

By means of the Hermitian metric and Chern connection, Qiu [4] obtained the Koppelman–Leray–Norguet formula for ( p, q) differential forms on an open set with C 1 piecewise smooth boundary on a Stein manifold, and under suitable conditions gave the solutions of ▪-equation on a Stein manifold. In this article, using the method of Range and Siu [5], under suitable conditions, the authors complicatedly calculate to give the uniform estimates of solutions of ▪-equation for ( p, q) differential forms on a Stein manifold.

An intrinsic volume functional on almost complex 6-manifolds and nearly Kähler geometry

Let (M, I) be an almost complex 6-manifold. The obstruction to the integrability of almost complex structure N:3 0,1 (M)!3 2,0 (M) (the so-called Nijenhuis tensor) maps one 3-dimensional bundle to another 3-dimensional bundle. We say that Nijenhuis tensor is nondegenerate if it is an isomorphism. An almost complex manifold (M, I) is called nearly Kahler if it admits a Hermitian form ! such that r(!) is totally antisymmetric, r being the Levi-Civita connection. We show that a nearly Kahler metric on a given almost complex 6-manifold with nondegenerate Nijenhuis tensor is unique (up to a constant). We interpret the nearly Kahler property in terms of G2geometry and in terms of connections with totally antisymmetric torsion, obtaining a number of equivalent definitions. Weconstructanaturaldiffeomorphism-invariantfunctional I ! R M VolI on the space of almost complex structures on M, similar to the Hitchin functional, and compute its extrema in the following important case. Consider an almost complex structure I with nondegenerate Nijenhuis tensor, admitting a Hermitian connection with totally antisymmetric torsion. We show that the Hitchin-like functional I ! R M VolI has an extremum in I if and only if (M, I) is nearly Kahler.

Formula omitted]-structures on submanifolds of a [formula omitted]-manifold

Local SU ( 3 ) -structures on an oriented submanifold of Spin ( 7 ) -manifold are determined and their types are characterized in terms of the shape operator and the type of Spin ( 7 ) -structure. An application to Bryant [R.L. Bryant, Submanifolds and special structures on the octonions, J. Diff. Geom. 17 (1982) 185–232] and Calabi [E. Calabi, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Amer. Math. Soc. 87 (1958) 407–438] examples is given. It is shown that the product of a Cayley plane and a minimal surface lying in a four-dimensional orthogonal Cayley plane with the induced complex structure from the octonions described by Bryant admits a holomorphic local complex volume form exactly when it lies in a three-plane, i.e. it coincides with the example constructed by Calabi. In this case the holomorphic ( 3 , 0 ) -form is parallel with respect to the unique Hermitian connection with totally skew-symmetric torsion.

Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic

We develop an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson. As illustrations, we construct numerically the hermitian Einstein metrics on the tangent bundle and a rank three vector bundle on P^2. In addition, we find a hermitian Yang-Mills connection on a stable rank three vector bundle on the Fermat quintic.