Hler Structure Research Articles

Left-invariant nearly pseudo-K\"{a}hler structures and the tangent Lie group

Left-invariant nearly pseudo-K\"{a}hler structures and the tangent Lie group

Hermitian non-K<span class="BxUVEf ILfuVd" lang="en"><span class="hgKElc">ä</span></span>hler structures on products of principal $\mathrm{S}^1$-bundles over complex flag manifolds and applications in Hermitian geometry with torsion

Hermitian non-K<span class="BxUVEf ILfuVd" lang="en"><span class="hgKElc">ä</span></span>hler structures on products of principal $\mathrm{S}^1$-bundles over complex flag manifolds and applications in Hermitian geometry with torsion

THE NON-DEGENERACY OF THE SKEW-SYMMETRIC BILINEAR FORM OF THE FINITE DIMENSIONAL REAL FROBENIUS LIE ALGEBRA

A Frobenius Lie algebra is recognized as the Lie algebra whose stabilizer at a Frobenius functional is trivial. This condition is equivalent to the existence of a skew-symmetric bilinear form which is non-degenerate. On the other hand, the Lie algebra is Frobenius as well if its orbit on the dual vector space is open. In this paper, we study the skew-symmetric bilinear form of finite dimensional Frobenius Lie algebra corresponding to its Frobenius functional. The work aims to prove that a Lie algebra of dimension is Frobenius if and only if the -th derivation of the Frobenius functional is not equal to zero. Indeed, this condition implies that the skew-symmetric bilinear form is non-degenerate and vice versa. In addition, some properties of Frobenius functionals are obtained. Furthermore, the computations are given using the coadjoint orbits and the structure matrix. As a discussion, we can investigate these results in the algebra case whether giving rise to a left-invariant K hler structure of a Frobenius Lie group or not.

Conformal changes of generalized paracontact pseudometric structures

We characterize generalized almost paracontact pseudometric structures after a conformal change and investigate invariant components of these structures under conformal changes. Then we characterize the conformal changes of normal generalized paracontact pseudometric structures, generalized almost para-co-K{\"a}hler structures, and generalized para-co-K{\"a}hler structures. We also give examples of a generalized almost para-co-K{\"a}hler structure and a normal generalized paracontact pseudometric structure that remains invariant under a nonhomothety conformal change.

Anti-Kählerian Geometry on Lie Groups

Let $G$ be a Lie group of even dimension and let $(g,J)$ be a left invariant anti-K\"ahler structure on $G$. In this article we study anti-K\"{a}hler structures considering the distinguished cases where the complex structure $J$ is abelian or bi-invariant. We find that if $G$ admits a left invariant anti-K\"ahler structure $(g,J)$ where $J$ is abelian then the Lie algebra of $G$ is unimodular and $(G,g)$ is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric $g$ for which $J$ is an anti-isometry we obtain that the triple $(G, g, J)$ is an anti-K\"ahler manifold. Besides, given a left invariant anti-Hermitian structure on $G$ we associate a covariant $3$-tensor $\theta$ on its Lie algebra and prove that such structure is anti-K\"ahler if and only if $\theta$ is a skew-symmetric and pure tensor. From this tensor we classify the real 4-dimensional Lie algebras for which the corresponding Lie group has a left invariant anti-K\"ahler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-K\"ahler structures).

Kähler structure in the commutative limit of matrix geometry

We consider the commutative limit of matrix geometry described by a large-$N$ sequence of some Hermitian matrices. Under some assumptions, we show that the commutative geometry possesses a K\"{a}hler structure. We find an explicit relation between the K\"{a}hler structure and the matrix configurations which define the matrix geometry. We also find a relation between the matrix configurations and those obtained from the geometric quantization.

Gauged linear sigma model with F-term for A-type ALE space

We construct yet another ${\mathcal N}=(4,4)$ gauged linear sigma model for the $A_N$-type ALE space. In our construction the toric data of the ALE space are manifest. Due to the $SU(2)_R$ symmetry, the F-term is automatically determined. The toric data, which govern the K\"{a}hler structures of the ALE space, are embedded into $U(1)$ charges of charged hypermultiplets. The F-term is also inevitable to determine the complex structures of the ALE space. In the IR limit, we obtain the K\"{a}hler potential of the $A_N$-type ALE space. We also find the origin of the ${\mathbb Z}_{N+1}$ orbifold symmetry in the singular limit of the $A_N$-type ALE space. In a special case, we reproduce an explicit form of the K\"{a}hler potential of the $A_1$-type ALE space, i.e., the Eguchi-Hanson space.

Riemannian Spin holonomy manifold carries octonionic-Kähler structure

We prove that Riemannian $Spin(7)$ holonomy manifolds carry octonionic-K\{a}hler structure.

Dimensional reduction of the heterotic string over nearly-Kähler manifolds

Our aim is to derive the effective action in four dimensions resulting by reducing dimensionally the ten-dimensional = 1 heterotic supergravity coupled to = 1 super Yang-Mills over manifolds admitting a nearly-K?hler structure. Given the fact that all homogeneous six-dimensional nearly-K?hler manifolds are included in the class of the corresponding non-symmetric coset spaces plus a group manifold, our procedure amounts in applying the Coset Space Dimensional Reduction scheme using these coset spaces as internal manifolds. In our examination firstly the rules of the reduction of the theory over a general six-dimensional non-symmetric manifold are stated and subsequently a detailed case by case analysis is performed for all the three non-symmetric coset spaces. For each case the four-dimensional scalar potential is derived and the corresponding nearly-K?hler limit is obtained. Finally, we determine the corresponding supergravity description of the four-dimensional theory employing the heterotic Gukov-Vafa-Witten formula and results of the special K?hler geometry.

Hilbert flag varieties and their K hler structure

In this paper, we introduce the infinite-dimensional flag varieties associated with integrable systems of the KdV- and Toda-type and discuss the structure of these manifolds. As an example we treat the Fubini–Study metric on the projective space associated with a separable complex Hilbert space and conclude by showing that all flag varieties introduced before possess a Kahler structure.

A K�hler structure on the moduli space of isometric maps of a circle into Euclidean space

We study the spaces ℳ and ℳ and ℳ′Lip of smooth (resp. non-degenerate Lipschitz) isometric maps of a circle into Euclidean space modulo orientation preserving Euclidean motions. We prove that ℳ and ℳ′Lip are infinite dimensional Kahler manifolds. In particular, they are complex Frechet (resp. Banach) manifolds. This is proved by an infinite dimensional version of the Kirwan, Kempf-Ness Theorem [Kir84], [KN78], [Nes84] relating symplectic quotients to holomorphic quotients, applied to the action ofPSL 2(ℂ) on the free loop space ofS 2.

Some simple examples of almost K�hler non-K�hler structures

Some simple examples of almost K�hler non-K�hler structures

Non-existence of almost K�hler structure on hyperbolic spaces of dimension2n(?4)

Non-existence of almost K�hler structure on hyperbolic spaces of dimension2n(?4)

The Weil-Petersson symplectic structure at Thurston’s boundary

The Weil-Petersson Kähler structure on the Teichmüller space $\mathcal {T}$ of a punctured surface is shown to extend, in an appropriate sense, to Thurston’s symplectic structure on the space $\mathcal {M}{\mathcal {F}_0}$ of measured foliations of compact support on the surface. We introduce a space ${\widetilde {\mathcal {M}\mathcal {F}}_0}$ of decorated measured foliations whose relationship to $\mathcal {M}{\mathcal {F}_0}$ is analogous to the relationship between the decorated Teichmüller space $\tilde {\mathcal {T}}$ and $\mathcal {T}$. $\widetilde {\mathcal {M}{\mathcal {F}_0}}$ is parametrized by a vector space, and there is a natural piecewise-linear embedding of $\mathcal {M}{\mathcal {F}_0}$ in $\widetilde {\mathcal {M}{\mathcal {F}_0}}$ which pulls back a global differential form to Thurston’s symplectic form. We exhibit a homeomorphism between $\tilde {\mathcal {T}}$ and ${\widetilde {\mathcal {M}\mathcal {F}}_0}$ which preserves the natural two-forms on these spaces. Following Thurston, we finally consider the space $\mathcal {Y}$ of all suitable classes of metrics of constant Gaussian curvature on the surface, form a natural completion $\overline {\mathcal {Y}}$ of $\mathcal {Y}$, and identify $\overline {\mathcal {Y}} - \mathcal {Y}$ with $\mathcal {M}{\mathcal {F}_0}$. An extension of the Weil-Petersson Kähler form to $\mathcal {Y}$ is found to extend continuously by Thurston’s symplectic pairing on $\mathcal {M}{\mathcal {F}_0}$ to a two-form on $\overline {\mathcal {Y}}$ itself.

K�hler structures on K-orbits of the group of diffeomorphisms of a circle

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Narasimhan?Seshadri connection and K�hler structure of the space of moduli of holomorphic vector bundles over Riemann surfaces

Narasimhan?Seshadri connection and K�hler structure of the space of moduli of holomorphic vector bundles over Riemann surfaces