Lie Groups Actions on Non Orientable Klein Surfaces

Bârză, Ilie; Ghişa, Dorin

doi:10.1007/978-981-15-7775-8_33

Cited by 4 publications

(4 citation statements)

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“…The m-Möbius transformations have been introduced in connection with Lie groups' actions on complex manifolds (see [3] and [4]). They represent an interesting mathematical topic in itself and we dedicated ourselves to performing in this paper a study of these transformations parallel to that of classical Möbius transformations of the complex plane.…”

Section: Discussionmentioning

confidence: 99%

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Multipliers and Classification of m-Möbius Transformations

Ghişa¹,

Mikulin²

2022

APM

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It is known that any m-Möbius transformation is an ordinary Möbius transformation in every one of its variables when the other variables do not take the values a and 1/a, where a is a parameter defining the respective m-Möbius transformation. For ordinary Möbius transformations having distinct fixed points, the multiplier associated with one of these points completely characterizes the nature of that transformation, i.e. it tells us if it is elliptic, hyperbolic or loxodromic. The purpose of this paper is to show that fixed points exist also for m-Möbius transformations and multipliers associated with them can be computed as well. As in the classical case, the values of those multipliers describe completely the nature of the transformations. The method we used was that of a thorough study of the coefficients of the variables involved, with which occasion we discovered surprising symmetries. These were the results allowing us to prove the main theorem regarding the fixed points of a m-Möbius transformation, which is the key to further developments. Finally we were able to illustrate the geometric aspects of these transformations, making the whole theory as intuitive as possible. It was as opening a window into a space of several complex variables. This allows us to prove that if a bi-Möbius transformation is elliptic or hyperbolic in z 1 at a point z 2 it will remain the same on a circle or line passing through z 2 . This property remains true when we switch z 1 and z 2 . The main theorem, dealing with the fixed points of an arbitrary m-Möbius transformation made possible the extension of this result to these transformations.

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Section: Discussionmentioning

confidence: 99%

“…The geometric properties of m-Möbius transformations revealed in [1] have been expanded in this paper by using the tool of multipliers. This became possible after proving that regarded as an ordi- The topic we dealt with here is a new one and it has been studied just in [1] [2] [3] [4]. No other reference was needed.…”

Section: Discussionmentioning

confidence: 99%

Multipliers and Classification of m-Möbius Transformations

Ghişa¹,

Mikulin²

2022

APM

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“…We dealt in [1] and [2] with Lie groups of bi-Möbius transformations defined on  . The concept can be extended linearly to n  in the following way.…”

Section: Introductionmentioning

confidence: 99%

Lie Groups Actions on Non Orientable n-Dimensional Complex Manifolds

Cao-Huu¹,

Ghişa²

2021

APM

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Analytic atlases on n can be easily defined making it an n-dimensional complex manifold. Then with the help of bi-Möbius transformations in complex coordinates Abelian groups are constructed making this manifold a Lie group. Actions of Lie groups on differentiable manifolds are well known and serve different purposes. We have introduced in previous works actions of Lie groups on non orientable Klein surfaces. The purpose of this work is to extend those studies to non orientable n-dimensional complex manifolds. Such manifolds are obtained by factorizing n  with the two elements group of a fixed point free antianalytic involution of n  . Involutions ( ) h z of this kind are obtained linearly by composing special Möbius transformations of the planes with the mapping ( ) 1. A convenient partition of n  is performed which helps defining an internal operation on n  h and finally actions of the previously defined Lie groups on the non orientable manifold n  h are displayed.

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“…When investigating Lie groups of Möbius transformations of the Riemann sphere, we were brought in [1] [2] and [3] to the study of some bi-Möbius transformations. These are functions : f × →    of the form:…”

Section: Introductionmentioning

confidence: 99%

A Note on m-Möbius Transformations

Ghişa¹

2021

APM

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Lie Groups Actions on Non Orientable Klein Surfaces

Cited by 4 publications

References 3 publications

Multipliers and Classification of <i>m</i>-Möbius Transformations

Multipliers and Classification of <i>m</i>-Möbius Transformations

Lie Groups Actions on Non Orientable <i>n</i>-Dimensional Complex Manifolds

A Note on <i>m</i>-Möbius Transformations

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Lie Groups Actions on Non Orientable Klein Surfaces

Cited by 4 publications

References 3 publications

Multipliers and Classification of &lt;i&gt;m&lt;/i&gt;-M&#246;bius Transformations

Multipliers and Classification of &lt;i&gt;m&lt;/i&gt;-M&#246;bius Transformations

Lie Groups Actions on Non Orientable &lt;i&gt;n&lt;/i&gt;-Dimensional Complex Manifolds

A Note on &lt;i&gt;m&lt;/i&gt;-M&#246;bius Transformations

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Multipliers and Classification of <i>m</i>-Möbius Transformations

Multipliers and Classification of <i>m</i>-Möbius Transformations

Lie Groups Actions on Non Orientable <i>n</i>-Dimensional Complex Manifolds

A Note on <i>m</i>-Möbius Transformations