On compact affine quaternionic curves and surfaces. (English) Zbl 1479.30035
This paper is written for specialists in complex and hypercomplex (quaternionic) geometry. It studies affine quaternionic manifolds, compact affine quaternionic curves and surfaces. Section 1 introduces affine quaternionic manifolds, especially for quaternionic dimensions 2 and 4. Section 2 first relates quaternionic and complex matrices and defines a quaternionic affine structure for \(4n\)-dimensional real differentiable manifolds. An important role is played by freely acting and properly discontinuous subgroups of quaternionic affine transformations over \(\mathbb{H}^n\), and their quotient spaces. It is found that an affine quaternionic manifold has one and only one affine quaternionic structure. Section 3 works towards the classification of affine quaternionic manifolds in low dimensions. Important are quaternionic right eigenvalues of \(n\times n\) quaternionic matrices. Further study is done for \(n=2\) subgroups of quaternionic affine transformations. A key result is Theorem 3.18 relating freely and properly discontinuously acting subgroups of quaternionic affine transformations (\(n=2\)) wih a unipotent normal subgroup of finite index, such that the quotient of the two is isomorphic to a finite subgroup of \(\mathbb{S}^3\), followed by an example. Finally, Section 4 treats the abelian case, where real, complex and quaternionic Heisenberg groups of matrices play an essential role.
Reviewer: Eckhard Hitzer (Tokyo)
MSC:
| 30G35 | Functions of hypercomplex variables and generalized variables |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |
Keywords:
affine quaternionic manifolds; fundamental groups of compact affine quaternionic surfaces; classification; quaternionic matrix; Heisenberg groupReferences:
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