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 defin...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<sub>1</sub> at a point z<sub>2</sub> it will remain the same on a circle or line passing through z<sub>2</sub>. This property remains true when we switch z<sub>1</sub> and z<sub>2</sub>. The main theorem, dealing with the fixed points of an arbitrary m-Möbius transformation made possible the extension of this result to these transformations.展开更多
Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius tran...Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations f<sub>m</sub> mapping onto . Even for the simplest entity, the pre-image by f<sub>m</sub> of a unique point, there is no way of visualization. Pre-images by f<sub>m</sub> of figures from C are like ghost figures in C<sup>m</sup>. This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most important achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in C<sup>m </sup>are well known and vector calculus in C<sup>m</sup> is familiar, yet the pre-image by f<sub>m</sub> of a vector from C is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in C<sup>m</sup>. Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points by those transformations translate into similar theorems for m-Möbius transformations.展开更多
Lie groups of bi-M<span style="white-space:nowrap;">ö</span>bius transformations are known and their actions on non orientable <em>n</em>-dimensional complex manifolds have b...Lie groups of bi-M<span style="white-space:nowrap;">ö</span>bius transformations are known and their actions on non orientable <em>n</em>-dimensional complex manifolds have been studied. In this paper, <em>m</em>-M<span style="white-space:nowrap;">ö</span>bius transformations are introduced and similar problems as those related to bi-M<span style="white-space:nowrap;">ö</span>bius transformations are tackled. In particular, it is shown that the subgroup generated by every <em>m</em>-M<span style="white-space:nowrap;">ö</span>bius transformation is a discrete group. Cyclic subgroups are exhibited. Vector valued <em>m</em>-M<span style="white-space:nowrap;">ö</span>bius transformations are also studied.展开更多
We shall prove the equivalences of a non-degenerate circle-preserving map and a M(o)bius transformation in ^Rn, of a non-degenerate geodesic-preserving map and an isometry in Hn of a non-degenerate line-preserving map...We shall prove the equivalences of a non-degenerate circle-preserving map and a M(o)bius transformation in ^Rn, of a non-degenerate geodesic-preserving map and an isometry in Hn of a non-degenerate line-preserving map and an affine transformation in Rn. That a map is non-degenerate means that the image of the whole space under the map is not a circle, or geodesic or line respectively. These results hold without either injective or surjective, or even continuous assumptions, which are new and of a fundamental nature in geometry.展开更多
Analytic atlases on <img src="Edit_948e45b7-cbef-425e-bb79-28648b859994.png" width="23" height="22" alt="" /> can be easily defined making it an <em>n</em>-dim...Analytic atlases on <img src="Edit_948e45b7-cbef-425e-bb79-28648b859994.png" width="23" height="22" alt="" /> can be easily defined making it an <em>n</em>-dimensional complex manifold. Then with the help of bi-M<span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span>bius 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 <em>n</em>-dimensional complex manifolds. Such manifolds are obtained by factorizing <img src="Edit_7e5721ee-bb7f-4224-bd52-7d4641ac1c73.png" width="23" height="22" alt="" /> with the two elements group of a fixed point free antianalytic involution of <img src="Edit_ddfdac64-b296-48c5-9bb2-932eb3d76008.png" width="23" height="22" alt="" />. Involutions <strong>h(z)</strong> of this kind are obtained linearly by composing special M<span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span>bius transformations of the planes with the mapping <img src="Edit_4cda269a-9399-41ae-a5da-0c9d18a419ab.png" width="89" height="24" alt="" /><img src="Edit_4cda269a-9399-41ae-a5da-0c9d18a419ab.png" width="85" height="22" alt="" />. A convenient partition of <img src="Edit_9e899708-41b0-4351-a12b-cc6efb5b1581.png" width="23" height="22" alt="" /> is performed which helps defining an internal operation on <img src="Edit_7cd42987-68f8-4e4c-9382-cbc68b56377e.png" width="49" height="26" alt="" /> and finally actions of the previously defined Lie groups on the non orientable manifold <img src="Edit_5740b48c-f9ea-438d-a87d-8cdc1f83662b.png" width="49" height="25" alt="" /> are displayed.展开更多
文摘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<sub>1</sub> at a point z<sub>2</sub> it will remain the same on a circle or line passing through z<sub>2</sub>. This property remains true when we switch z<sub>1</sub> and z<sub>2</sub>. The main theorem, dealing with the fixed points of an arbitrary m-Möbius transformation made possible the extension of this result to these transformations.
文摘Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations f<sub>m</sub> mapping onto . Even for the simplest entity, the pre-image by f<sub>m</sub> of a unique point, there is no way of visualization. Pre-images by f<sub>m</sub> of figures from C are like ghost figures in C<sup>m</sup>. This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most important achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in C<sup>m </sup>are well known and vector calculus in C<sup>m</sup> is familiar, yet the pre-image by f<sub>m</sub> of a vector from C is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in C<sup>m</sup>. Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points by those transformations translate into similar theorems for m-Möbius transformations.
文摘Lie groups of bi-M<span style="white-space:nowrap;">ö</span>bius transformations are known and their actions on non orientable <em>n</em>-dimensional complex manifolds have been studied. In this paper, <em>m</em>-M<span style="white-space:nowrap;">ö</span>bius transformations are introduced and similar problems as those related to bi-M<span style="white-space:nowrap;">ö</span>bius transformations are tackled. In particular, it is shown that the subgroup generated by every <em>m</em>-M<span style="white-space:nowrap;">ö</span>bius transformation is a discrete group. Cyclic subgroups are exhibited. Vector valued <em>m</em>-M<span style="white-space:nowrap;">ö</span>bius transformations are also studied.
基金the National Natural Science Foundation of China(Grant No.10125103)the 973 Project of China.
文摘We shall prove the equivalences of a non-degenerate circle-preserving map and a M(o)bius transformation in ^Rn, of a non-degenerate geodesic-preserving map and an isometry in Hn of a non-degenerate line-preserving map and an affine transformation in Rn. That a map is non-degenerate means that the image of the whole space under the map is not a circle, or geodesic or line respectively. These results hold without either injective or surjective, or even continuous assumptions, which are new and of a fundamental nature in geometry.
文摘Analytic atlases on <img src="Edit_948e45b7-cbef-425e-bb79-28648b859994.png" width="23" height="22" alt="" /> can be easily defined making it an <em>n</em>-dimensional complex manifold. Then with the help of bi-M<span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span>bius 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 <em>n</em>-dimensional complex manifolds. Such manifolds are obtained by factorizing <img src="Edit_7e5721ee-bb7f-4224-bd52-7d4641ac1c73.png" width="23" height="22" alt="" /> with the two elements group of a fixed point free antianalytic involution of <img src="Edit_ddfdac64-b296-48c5-9bb2-932eb3d76008.png" width="23" height="22" alt="" />. Involutions <strong>h(z)</strong> of this kind are obtained linearly by composing special M<span style="white-space:nowrap;"><span style="white-space:nowrap;">ö</span></span>bius transformations of the planes with the mapping <img src="Edit_4cda269a-9399-41ae-a5da-0c9d18a419ab.png" width="89" height="24" alt="" /><img src="Edit_4cda269a-9399-41ae-a5da-0c9d18a419ab.png" width="85" height="22" alt="" />. A convenient partition of <img src="Edit_9e899708-41b0-4351-a12b-cc6efb5b1581.png" width="23" height="22" alt="" /> is performed which helps defining an internal operation on <img src="Edit_7cd42987-68f8-4e4c-9382-cbc68b56377e.png" width="49" height="26" alt="" /> and finally actions of the previously defined Lie groups on the non orientable manifold <img src="Edit_5740b48c-f9ea-438d-a87d-8cdc1f83662b.png" width="49" height="25" alt="" /> are displayed.
