Yat Sun Poon

Conformal normal curvature and assessment of local influence

In 1986, R. D. Cook proposed differential geometry to assess local influence of minor perturbations of statistical models. We construct a conformally invariant curvature, the conformal normal curvature, for the same purpose. This curvature provides a measure of local influence ranging from 0 to 1, with objective bench-marks to judge largeness. We study various approaches to using the conformal normal curvature and the relationships between these approaches.

Geometry of Hyper-Kähler Connections with Torsion

Abstract:The internal space of a N = 4 supersymmetric model with Wess–Zumino term has a connection with totally skew-symmetric torsion and holonomy in SP(n). We study the mathematical background of this type of connection. In particular, we relate it to classical Hermitian geometry, construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory.

Hyper-Kähler metrics and a generalization of the Bogomolny equations

We generalize the Bogomolny equations to field equations on ℝ3 ⊘ ℝn and describe a twistor correspondence. We consider a general hyper-Kähler metric in dimension 4n with an action of the torusTn compatible with the hyper-Kähler structure. We prove that such a metric can be described in terms of theTn-solution of the field equations coming from the twistor space of the metric.

The Einstein-Weyl equations in complex and quaternionic geometry*

Abstract Einstein–Weyl manifolds with compatible complex structures are shown to be given as torus bundles on Kahler–Einstein manifolds, extending known results on locally conformal Kahler manifolds. The Weyl structure is derived from a Ricci-flat metric constructed by Calabi on the canonical bundle of the Kahler–Einstein manifold. Similar questions are addressed when the Weyl geometry admits compatible hypercomplex or quaternionic structures.

Deformation of 2-Step Nilmanifolds with Abelian Complex Structures

We develop deformation theory for abelian invariant complex structures on a nilmanifold, and prove that in this case the invariance property is preserved by the Kuranishi process. A purely algebraic condition characterizes the deformations leading again to abelian structures, and we prove that such deformations are unobstructed. Various examples illustrate the resulting theory, and the behavior possible in three complex dimensions.

STABILITY OF ABELIAN COMPLEX STRUCTURES

Let M =Γ \G be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result in [7] for 2-step nilmanifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the deformation process of abelian complex structures is stable within the class of nilpotent complex structures. We give an example to show that this property does not hold in higher dimension.

Hypercomplex structures associated to quaternionic manifolds

Abstract If M is a quaternionic manifold and P is an S1-instanton over M, then Joyce constructed a hypercomplex manifold we call ℘(M) over M. These hypercomplex manifolds admit a U(2)-action of a special type permuting the complex structures. We show that up to double covers, all such hypercomplex manifolds arise in this way. Examples, including that of a hypercomplex structure on SU(3), show the necessity of including double covers of ℘(M) .

Duality and Yang-Mills fields on quaternionic Kähler manifolds

Electronic tuning apparatus comprises an electronic tuner capable of tuning in a plurality of channels within a predetermined frequency range in response to a variable tuning signal, a device for selectively generating a channel selecting signal corresponding to a desired channel from among the plurality of channels, a storage device for storing a number of predetermined tuning values, said number being less than the total number of the plurality of channels and said storage device having an output to which the predetermined tuning values are selectively delivered, an operational processor for generating an actual tuning value for tuning to the desired channel in response to said channel selecting signal and the output of the storage device, and a device which receives the actual tuning value from the operational processor for supplying a corresponding tuning signal to the electronic tuner.

Kahler surfaces with zero scalar curvature

The equations of Euler type used by Belinskii et al. (1978) to generate Ricci-flat Kahler metrics on 4-space are generalized to be a system of equations that generates Kahler metrics with vanishing scalar curvature. This family of new metrics also contains a family of scalar-flat Kahler surfaces constructed by LeBrun (1988).

Generalized contact structures

We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odddimensional manifolds. We name the latter strong generalized contact structures. Using a Boothby-Wang construction bridging symplectic structures and contact structures, we find examples to demonstrate that, within the category of generalized contact structures, classical contact structures have non-trivial deformations. Using deformation theory of Lie bialgebroids, we construct new families of strong generalized contact structures on the threedimensional Heisenberg group and its co-compact quotients. Address: Department of Mathematics, University of California at Riverside, Riverside CA 92521, U.S.A., Email: ypoon@ucr.edu. Partially supported by UCMEXUS and NSF-0906264 Address: Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A., Email: wade@math.psu.edu