arXiv math/99111211999-11-16math.DGgr-qcmath-phmath.MPnlin.SIsolv-int

Einstein metrics, hypercomplex structures and the Toda field equation

David M. J. Calderbank, Paul Tod

备注：: 9 pages
发表信息：: Differ.Geom.Appl. 14 (2001) 199-208

英文摘要：

We obtain explicitly all solutions of the SU(infinity) Toda field equation with the property that the associated Einstein-Weyl space admits a 2-sphere of divergence-free shear-free geodesic congruences. The solutions depend on an arbitrary holomorphic function and give rise to new hyperKahler and selfdual Einstein metrics with one dimensional isometry group. These metrics each admit a compatible hypercomplex structure with respect to which the symmetries are triholomorphic.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

关于同构HKT流形与爱因斯坦条件

On homogeneous HKT manifolds and the Einstein condition

Università dell’Aquila（阿奎拉大学）

研究了具有紧致李群传递作用的同构超复流形，给出了不变HKT度量的特征，并证明了存在唯一的标度不变HKT-爱因斯坦度量，同时确定了哪些不变HKT度量的Bismut联络的 torsion 和 curvature 是Bismut-平行的。

2026-04-24math.DG

超复几何中的特殊度量

Special metrics in hypercomplex geometry

本文研究特殊超复度量的存在性和几何性质，探讨了四元数Gauduchon度量与超复上同调不变量消失的条件，证明了强HKT与平衡超复度量的不相容性，并引入了一种爱因斯坦型条件。

2026-04-24math.CVmath.DG

Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics

We study the Jones and Tod correspondence between selfdual conformal 4-manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3-manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kahler metrics and hypercomplex structures with symmetry. We also show that in the presence of such a congruence, the Einstein-Weyl equation is equivalent to a pair of coupled monopole eq…

1999-11-16math.AGmath.DG

Selfdual Einstein metrics and conformal submersions

Weyl derivatives, Weyl-Lie derivatives and conformal submersions are defined, then used to generalize the Jones-Tod correspondence between selfdual 4-manifolds with symmetry and Einstein-Weyl 3-manifolds with an abelian monopole. In this generalization, the conformal symmetry is replaced by a particular kind of conformal submersion with one dimensional fibres. Special cases are studied in which the conformal submersion is holomorphic, affine, or projective. All scalar-flat Kahler metrics with such a holomorphic conformal submersion, and all four dimensional hypercomplex structures with a compa…

2000-01-07gr-qcmath-phmath.DG

The geometry of the Toda equation

I show that solutions of the SU(infinity) Toda field equation generating a fixed Einstein-Weyl space are governed by a linear equation on the Einstein-Weyl space. From this, obstructions to the existence of Toda solutions generating a given Einstein-Weyl space are found. I also give a classification of Einstein-Weyl spaces arising from the Toda equation in more than one way. This classification coincides with a class of spaces found by Ward and hence clarifies some of their properties. I end by discussing the simplest examples.

2000-01-26math-phmath.DGmath.MP

Einstein Metrics, Projective Structures and the $SU(\infty)$ Toda Equation

We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(\infty)$ Toda equation. We give several examples of new, explicit solutions of the Toda equation, and construct their mini--twistor spaces. Finally we discuss the projective-to-Einstein correspondence, which gives a neutral signature Einstein metric on a cotangent bundle $T^*N$ of any projective structure $(N, [\nabla])$. We show that there is a canonical Einstein of metric on an $\R^*$--bundle over $T^*N$, with a connection whose curva…

2019-11-05gr-qchep-thmath.DG

更多相关论文收起更多论文

超Kähler流形上丛的Einstein度量

Einstein metrics on bundles over hyperKähler manifolds

本文在超Kähler流形上的丛中构造显式四元数-Kähler与超复结构，借助Galicki-Lawson矩映射进行QK约化，得到显式度量并给出超Kähler/四元数-Kähler对应的新证明及其他Einstein、平衡Hermitian结构例子。

2022-07-09math.DG

关于平衡HKT流形

On balanced HKT manifolds

本文证明了紧致超复流形上平衡HKT锥的开放性，研究了超全纯向量场的李代数及其与平衡HKT结构的相互作用，并证明了某些流形上不存在此类向量场。

2025-06-18math.DG

Quaternionic Kaehler manifolds with Hermitian and Norden metrics

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.

2009-06-27math.DG

$SU(\infty)$-Toda 方程的边值问题：基于 Onofri 不等式

Boundary Value Problems for the $SU(\infty)$-Toda Equation via the Onofri Inequality

本文通过 Onofri 不等式解决了 $SU(\infty)$-Toda 方程的退化椭圆边值问题，构造了具有圆对称性的反自对偶庞加莱-爱因斯坦度量。

2026-09-30math-phmath.APmath.DG

具有共形Kähler几何的Poincaré-Einstein 4-流形

Poincaré-Einstein 4-manifolds with conformally Kähler geometry

本文研究共形类含Kähler度量的4维Poincaré-Einstein流形，将Einstein方程约化为Toda型方程，针对$\u212d^1$作用情形构建Dirichlet边值问题并证明存在唯一性，非微扰构造出共形无穷远为非正Yamabe型的无穷多族新度量。

2025-10-06math-phmath.APmath.DG

One-sided type-D Ricci-flat metrics

We consider four-dimensional, Riemannian, Ricci-flat metrics for which one or other of the self-dual or anti-self-dual Weyl tensors is type-D. Such metrics always have a valence-2 Killing spinor, and therefore a Hermitian structure and at least one Killing vector. We rederive the results of Przanowski and collaborators, that these metrics can all be given in terms of a solution of the $SU(\infty)$-Toda field equation, and show that, when there is a second Killing vector commuting with the first, the method of Ward can be applied to show that the metrics can also be given in terms of an axisymm…

2021-10-27gr-qcmath.DG

广义引力瞬子的隐藏对称性

Hidden symmetries of generalised gravitational instantons

提出一个框架，用于求解具有Killing场的共形Kähler黎曼四流形上对应共形自对偶和宇宙学Einstein-Maxwell的广义引力瞬子场方程，推导曲率恒等式，获得三类几何的SU(∞)Toda表述并求解，找到其广义化。

2024-11-11gr-qchep-thmath.DG

具有对齐Einstein-Maxwell的单边type-D度规

One-sided type-D metrics with aligned Einstein-Maxwell

研究四维Riemannian度规下type-D Weyl张量与Maxwell场对齐的Einstein-Maxwell方程，证明其可由$SU(\infty)$-Toda场方程或三维Laplacian的轴对称解表示，从而实现场方程线性化。

2024-10-17gr-qcmath.DG

Airy, Beltrami, Maxwell, Morera, Einstein and Lanczos potentials revisited

The main purpose of this paper is to revisit the well known potentials, called stress functions, needed in order to study the parametrizations of the stress equations, respectively provided by G.B. Airy (1863) for 2-dimensional elasticity, then by E. Beltrami (1892), J.C. Maxwell (1870) and G. Morera (1892) for 3-dimensional elasticity, finally by A. Einstein (1915) for 4-dimensional elasticity, both with a variational procedure introduced by C. Lanczos (1949,1962) in order to relate potentials to Lagrange multipliers. Using the methods of Algebraic Analysis, namely mixing differential geometr…

2015-12-18math-phmath.APmath.DG