半（次）黎曼流形上的共形和射影映射的几何不变性研究
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摘要
半黎曼流形是指赋予非退化约束的度量的微分流形.黎曼流形可视为半黎曼流形的一种特殊情况.半黎曼流形中另一个重要的情形就是Lorentz流形.广义相对论时空就是一个4维连通时间定向的Lorentz空间.1905年,Einstein认识到时空中的引力实际上就是半黎曼几何中的曲率,这一认识大大推动了半黎曼几何在相对论中的深入而广泛的应用,譬如半黎曼几何理论广泛渗透到宇宙学(红移、宇宙膨胀、宇宙大爆炸),单恒星引力(近日点进动、光线弯曲、黑洞)等领域的研究.
次黎曼流形,粗略地讲,就是被赋予了一个分布及此分布上的一个纤维内积的流形,当考虑的分布为整个切丛时,次黎曼流形就成为黎曼流形.次黎曼流形作为黎曼流形的一个自然发展,是度量空间上几何分析的研究的基本空间,也为次椭圆算子的分析和Cauchy-Riemannian流形的探究提供一个可能的共同的研究框架；另一方面,其在控制论、经典力学、规范场论和量子物理等方面也有着极其重要的理论和实际应用.
半、次黎曼几何的广泛应用也大大促进了此类几何空间本身的发展,使得黎曼几何中许多重要的结果都推广到了半、次黎曼流形中,如次黎曼空间之子流形几何、正规测地线、次射影映照、次共形映照等等研究.
对于射影对应和共形对应,人们通过这种对应的等价关系,分别构造出射影和共形不变量.利用对此类不变性的研究,实现了几何对象的分类和同化问题.如：常曲率度量的射影等价度量只能是常曲率的；完备的Einstein度量的射影等价度量只能是成比例的；闭的连通的黎曼流形上的连通射影映射群有了一个完全分类：其只能是等距群或者该流形能被球面覆盖；两个黎曼度量的射影等价性恰好约化成切丛上BM-结构的存在性.
众所周知,n维拉普拉斯方程的对称群恰好是欧氏空间中的共形映射群,利用共形对称性,可以将调和函数的定义推广到共形平坦的黎曼流形.轴对称度规可以描述某类不带电的旋转星球的外部时空几何等等.对于二维共形变换群情形,人们通过全纯函数理解共形特性,并构建了复分析中深刻的黎曼映射定理.
在微分几何中,Klein关于黎曼空间中的变换论的研究,构成了空间几何学分类的形成.共形平坦空间之常曲率空间研究,催生了Einstein流形定义的形成.再如等距映射保持截曲率不变,但反之未必成立.如加上测地线的限制PJacobi场的特性,则逆命题成立.这就是著名的Cartan等距定理.该定理的大范围推广是Ambrose和Hicks给出的,即Cartan-Ambrose-Hicks定理.
如上所述,变换论的研究对于揭示空间的几何特征,物理属性等,具有举足轻重的作用,有时甚至于是本质性的.继续开展这方面的深入研究也是很有必要的.在黎曼几何中,射影映照和共形映照都是微分几何中非常重要的学科分支,变换论的研究已经有了成熟的理论体系,但是在半、次黎曼几何中尚不完善,例如：是否还存在与共形变换有重大关系的半-Weyl共形曲率张量?拥有保圆的半共形变换流形上的测地线有何特性?次黎曼空间中保曲面次共形变换是否与其第一基本形式成比例?半黎曼几何中的广义对称空间是否在射影映照下具有不变性?广义Einstein是否是共形不变的?再如,次黎曼空间中的共形映照下的不变量以及射影等价联络问题等等都尚未彻底解决.这些问题都是很基本的问题,也是很有意义的问题.弄清这些问题,对于深入理解半、次黎曼流形的几何特性,开展此类空间中的诸如Killing场论的进一步研究,以及度规的内蕴“弯曲性”及运动不变性等等,是十分重要和迫切的.
本文拟主要研究半、次黎曼流形上射影和共形映照下的几何不变性及相关的应用问题.
文章的第一部分包括第二章和第三章,第二章讨论了伪对称半黎曼流形上的射影映照的性质.对具有半对称联络的伪对称半黎曼流形,给出了其上关于射影映照的一整体结果：伪对称的半黎曼流形关于射影映照形成了一个闭类;同时给出了伪对称半黎曼流形上的射影等价度量的分类.
第三章研究了半黎曼流形上保持时空模型广义拟Einstein空间不变的一类共形映照,在此类共形映照下,我们得到了一类具有几何特征的空间曲线-椭圆是保持不变的,称之为ξ,η-拟保圆映照,且得出该类映照能够把动力系统中的椭圆动力系变到椭圆动力系.此外,还得到了ξ,,η-拟保圆映照下的不变量,根据此不变量的性质,我们得出ξ,η-拟保圆映照的保持广义拟Einstein空间的不变性；另外,我们给出了容有ξ,η-拟保圆映照的流形具有的几何特征,特别的,研究了一大类容有ξ,η-拟保圆映照的循环流形的几何结构,包括Ruse循环、共圆循环、共调和循环、共形循环空间,得出容有ξ,η拟保圆映照的这一大类循环流形都约化为拟Einstein空间.
文章第二部分即第四章,主要研究次黎曼流形中的次共形映照下的不变性问题以及射影等价的非完整联络的特性.给出了“水平Einstein"(?)口“水平拟Einstein"空间的定义,并刻画了两种空间的几何特征.
对于次黎曼空间中的次共形映照,首先,找到了次共形映照下的几类几何不变量;
其次,研究并给出了次共形循环次黎曼流形约化为次共形平坦的次黎曼流形的充要条件以及次共形映照保持“水平Einstein"空间不变的充要条件；
最后,研究了次黎曼流形上的射影等价的非完整联络,给出了次黎曼流形上两个不同的非完整联络具有相同测地线的充要条件.
本文获得的主要结论如下：对于伪对称半黎曼空间上的射影映照我们有
定理2.2.1假设Ψ：(M,g)→(丽,动是从伪对称半黎曼空间(M,g)到半黎曼空间(M,g)的半对称射影映照,则半黎曼空间(M,g)也是伪对称的.
对于ξ,η-拟保圆映照,我们有
定理3.2.1如果两个半黎曼流形(M,g)和(M,g)成共形对应g=ρ2g,则该共形映照是ξ,η-拟保圆映照的充要条件是张量G或者其缩并张量G是不变的.
同时,ξ,η-拟保圆映照可以保持某些具有几何特征的空间曲线不变.
定理3.3.2如果在半黎曼流形M、M存在共形映照gij=ρ2gij,且函数ρ满足下面方程其中a，b,c是M上的任意函数.则在这类共形映照下,M中每个切方向为Mij的主方向的椭圆都变到M中的椭圆.
对于次黎曼流形中的次共形映照,我们找到了该映照下的几类几何不变量：水平托马斯联络系数、次Weyl共形曲率张量以及次共形第二基本张量,将文中定理4.2.1,4.2.2,4.2.3统一起来可叙述如下
定理在次共形映照下，次黎曼流形的水平托马斯联络系数、次Weyl共形曲率张量、次共形第二基本张量都是不变量.
特别地,由次共形第二基本张量的共形不变性,我们得到如下的
定理4.2.5在次黎曼流形的次共形变换下，全水平脐点子流形变为全水平脐点子流形.
对于次黎曼流形上的两个不同的非完整联络所对应的测地线之间的关系研究,得到了下面的
定理4.3.2假设▽,▽是次黎曼流形M上两个非完整联络,则下面结论是等价的
(1)▽和▽有相同的非完整测地线(不同的参数);
(2)对每一个X∈H,存在λx使得D(X,X)=λxX；
(3)存在唯一的1-形式ω使得S(X,Y)=ω(X)Y+ω(Y)X.
A semi-Riemannian manifold is a smooth manifold associated with a non-degenerate met-ric tensor. It is obvious that the Riemannian manifold is a special case of semi-Riemannian manifolds. A remarkable case of semi-Riemannian manifolds is the so-called Lorentz man-ifold. The general relativity spacetime is an exactly connected time-oriented4-dimensional Lorentz space.1905, Einstein had recognized that the gravitation in spacetime is in correspon-dence with the curvature in a semi-Riemannian space, which promotes the thread applications of semi-Riemannian geometry to the special and the general relativity, for cosmology (red-shift, expanding universe, and big bang) and the gravitation of a single star (perihelion precession, bending of light, and black holes).
A sub-Riemannian manifold, roughly speaking, is a smooth manifold associated with a distribution and a fibre-inner product on it. When the distribution is the whole tangent bundle, then the sub-Riemannian manifold reduces to be a Riemannian manifold. On the one hand, as a natural generalization of Riemannian spaces, the sub-Riemannian manifolds constitute the basic spaces for the geometry and analysis in metric spaces, and also provide a possible common research framework for the study of sub-elliptic operators and Cauchy-Riemannian manifolds; On the other hand, the sub-Riemannian manifolds have extensively been important theory and practical applications in control theory, classical mechanics, gauge field theory and quantum physics.
The increasingly extensive applications of semi-and sub-Riemannian manifolds also have contributed significantly to the development of the geometrical spaces, thus many important results are extended to semi-and sub-Riemannian manifolds, such as submanifold geometry of sub-Riemannian manifolds, normal geodesics, sub-projective mappings, sub-conformal map-pings and so on.
Projective and conformal mappings are all equivalent relations, through them one can con-struct projective and conformal invariants, respectively. And we can study the geometric ob-jects classification and identification by the invariants. For instance, the projective equivalent metrics of constant curvature metrics are also of constant curvature; and the projective equiv- alent metrics of complete Einstein metrics are proportional; a connected Lie group acts on a closed connected Riemannian manifold by transformations that preserve the geodesics (as non-parametric curves), then its acts by isometries or the manifold is covered by the round spheres; the projective equivalence of two metrics is reduced to the existence of BM-structures.
As is well known, the symmetry group of n-dimensional Laplace equation is exactly the conformal group in Euclidean space, using the conformal symmetry, one can extend the har-monic functions to the Riemannian manifolds of conformally flat. Axial symmetric metrics can describe the external geometry of spacetime of some class of uncharged rotating planets and so on. For the case of two-dimensional conformal transformation group, one can comprehend the conformal property via the holomorphic functions, and construct the profound Riemannian mapping theorem in complex analysis.
In differential geometry, the research of the transform theories posed by Klein forms the geometric classification of Riemannian spaces. The spaces of constant curvature are particularly important cases of conformally flat Riemannian manifolds, and the research of the spaces of constant curvature leads to the birth of Einstein manifold. Moreover, we know that an isometric mapping keeps the sectional curvature unchanged, but the inverse is not true. If there have the additional restrictions on the geodesics, then inverse is true. This is the famous Cartan isometric theorem. The global generalization is given by Ambrose and Hicks, i.e. Cartan-Ambrose-Hicks theorem.
As mentioned above, the transform theory plays a decisive and substantial role for the re-veal of the geometric characteristics, physical properties of the spaces. Thus it is necessary to continue to carry out the research in this field. In Riemannian geometry, projective mappings and conformal mappings are both important in differential geometry, and the research of them forms a comprehensive theory system, but in the semi-and sub-Riemannian geometry, there are some fundamental problems open. Such as whether there exists the semi-Weyl conformal curvature tensor related to the conformal transformations? Is it also true that? What is the properties of geodesics on the semi-Riemannian manifolds admitting concircular transforma-tion? Whether the first fundamental form of a surface is proportional under a sub-conformal transformation? And whether the generalized symmetric spaces are invariant under the projec- tive mapping and the generalized Einstein spaces are unchanged under the conformal mappings? Moreover, the conformal invariants and the projective equivalent connection in sub-Riemannian spaces are not resolved completely. These questions are basic, but of significant meanings. Thus it is very important to make clear all the problems mentioned above, which can leads to further understanding of the geometric characteristics of semi-and sub-Riemannian manifolds, and the further research of Killing field, the intrinsic bending, the invariance under the motions and so on.
This paper is concentrated on the invariance study of conformal and projective mappings in semi-and sub-Riemannian manifolds and their applications.
This paper consists of two parts, and is separated into four chapters. The first part contains the second and the third chapters. In the second chapter, we discuss the projective mappings of semi-Riemannian manifolds, especial the pseudo-symmetric semi-Riemannian manifolds admitting a semi-symmetric connection. We obtain a global result:The pseudo-symmetric semi-Riemannian manifolds form a closed class with respect to projective mappings, more-over, we give the classification of the projective equivalent metrics on pseudo-symmetric semi-Riemannian manifolds.
In the third chapter, we investigate a class of conformal mappings of semi-Riemannian manifolds which can keep generalized quasi-Einstein manifolds unchanged. We show that every ellipse in semi-Riemannian manifolds is transformed into a ellipse under this class of confor-mal mappings, so we call this type of conformal mappings the ξ,η-quasi concircular mappings. Moreover, we obtain that this class of conformal mappings take elliptic dynamical systems into elliptic dynamical systems. We also obtain the corresponding invariant under such a mapping, by means of the properties of the invariants, we get an ξ,η-quasi concircular mapping keeps a generalized quasi-Einstein manifold unchanged. Then we characterize the geometrical proper-ties of the semi-Riemannian manifolds admitting ξ,η-quasi concircular mappings, in particular, we study a big class of recurrent manifolds, including Ruse recurrent, concircularly recurrent, conharmonically recurrent, conformally recurrent, we prove that this big class of recurrent man-ifolds admitting ξ,η-quasi concircular mappings are all reduced to quasi-Einstein spaces.
The second part of the paper is exact the fourth chapter, we mainly study the geometrical invariance of sub-conformal mappings and the projective equivalent nonholonomic connections in sub-Riemannian manifolds. We introduce the horizontal Einstein space and the horizontal quasi-Einstein space and explain the geometrical characteristics of the two spaces. We also discuss the relation between the horizontal quasi-Einstein spaces and some symmetric sub-Riemannian manifolds. For the sub-conformal mappings of sub-Riemannian manifolds, we obtain several classes of invariants under the sub-conformal mappings, then we give a necessary and sufficient conditions of a sub-conformally recurrent sub-Riemannian manifold reducing to a sub-conformally symmetric space and a sub-conformal mapping keeping a horizontal Einstein space invariant. In the last, we discuss when two different nonholonomic connections on a sub-Riemannian manifold have the same geodesics, we give the necessary and sufficient conditions of two different nonholonomic connections have the same geodesics.
The main results of this paper are as follows:
For the projective mappings of pseudo-symmetric semi-Riemannian manifolds we have
Theorem2.2.1IfΨ:(M,g)→(M,g) is the semi-symmetric projective mapping from pseudo-symmetric semi-Riemannian manifold (M, g) to a semi-Riemannian manifold (M,g), then (M,g) is also a psendo-symmetric manifold.
For ξ,η-quasi concircular mappings, we get:
Theorem3.2.1Suppose that there exists an conformal mapping g=ρ2g between two semi-Riemannian manifolds (M,g) and (M,g), then, the conformal mapping is ξ, η-quasi concircular if and only if the tensor G or the contracted tensor G are invariants.
Moreover, we find there are the corresponding curves having some geometrical character-istics under this mapping:
Theorem3.3.2If there is a conformal mapping gij=ρ2gij between two semi-Riemannian manifolds M and M, and the function p satisfies the following equations where a, b, c are any scalars on M. Then, under this mapping, every ellipse in M with tangent direction being the principle direction of Mij is transformed to a ellipse in M.
For the sub-conformal mappings in sub-Riemannian manifolds, we find several classes of sub-conformal invariants:horizontal Thomas coefficients of connections, Weyl sub-conformal curvature tensor, the sub-conformal second tensor. From theorem4.2.1,4.2.2,4.2.3we have Theorem Under the sub-conformal mappings, the horizontal Thomas coefficients of connec-tion, the Weyl sub-conformal curvature tensor, the sub-conformal second tensor are all invari-ants.
In particular, in view of the conformal invariance of sub-conformal second tensor, we get
Theorem4.2.4The sub-conformal mappings take horizontal totally umbilical submanifolds to horizontal totally umbilical submanifolds.
At the last of the paper, we study the necessary and sufficient conditions of two different nonholonomic connections having the same geodesics.
Theorem4.3.2Suppose (?)and (?)are two nonholonomic connections on sub-Riemannian mani-fold M, then the following are equivalent
(1)(?) and (?)have the same geodesics (with the different parameters);
(2)for every X∈H, there exists λx such that D(X, X)=λXX;
(3)there exist a unique1-form ω such that S(X,Y)=ω(X)Y+ω(Y)X.
次黎曼流形,粗略地讲,就是被赋予了一个分布及此分布上的一个纤维内积的流形,当考虑的分布为整个切丛时,次黎曼流形就成为黎曼流形.次黎曼流形作为黎曼流形的一个自然发展,是度量空间上几何分析的研究的基本空间,也为次椭圆算子的分析和Cauchy-Riemannian流形的探究提供一个可能的共同的研究框架；另一方面,其在控制论、经典力学、规范场论和量子物理等方面也有着极其重要的理论和实际应用.
半、次黎曼几何的广泛应用也大大促进了此类几何空间本身的发展,使得黎曼几何中许多重要的结果都推广到了半、次黎曼流形中,如次黎曼空间之子流形几何、正规测地线、次射影映照、次共形映照等等研究.
对于射影对应和共形对应,人们通过这种对应的等价关系,分别构造出射影和共形不变量.利用对此类不变性的研究,实现了几何对象的分类和同化问题.如：常曲率度量的射影等价度量只能是常曲率的；完备的Einstein度量的射影等价度量只能是成比例的；闭的连通的黎曼流形上的连通射影映射群有了一个完全分类：其只能是等距群或者该流形能被球面覆盖；两个黎曼度量的射影等价性恰好约化成切丛上BM-结构的存在性.
众所周知,n维拉普拉斯方程的对称群恰好是欧氏空间中的共形映射群,利用共形对称性,可以将调和函数的定义推广到共形平坦的黎曼流形.轴对称度规可以描述某类不带电的旋转星球的外部时空几何等等.对于二维共形变换群情形,人们通过全纯函数理解共形特性,并构建了复分析中深刻的黎曼映射定理.
在微分几何中,Klein关于黎曼空间中的变换论的研究,构成了空间几何学分类的形成.共形平坦空间之常曲率空间研究,催生了Einstein流形定义的形成.再如等距映射保持截曲率不变,但反之未必成立.如加上测地线的限制PJacobi场的特性,则逆命题成立.这就是著名的Cartan等距定理.该定理的大范围推广是Ambrose和Hicks给出的,即Cartan-Ambrose-Hicks定理.
如上所述,变换论的研究对于揭示空间的几何特征,物理属性等,具有举足轻重的作用,有时甚至于是本质性的.继续开展这方面的深入研究也是很有必要的.在黎曼几何中,射影映照和共形映照都是微分几何中非常重要的学科分支,变换论的研究已经有了成熟的理论体系,但是在半、次黎曼几何中尚不完善,例如：是否还存在与共形变换有重大关系的半-Weyl共形曲率张量?拥有保圆的半共形变换流形上的测地线有何特性?次黎曼空间中保曲面次共形变换是否与其第一基本形式成比例?半黎曼几何中的广义对称空间是否在射影映照下具有不变性?广义Einstein是否是共形不变的?再如,次黎曼空间中的共形映照下的不变量以及射影等价联络问题等等都尚未彻底解决.这些问题都是很基本的问题,也是很有意义的问题.弄清这些问题,对于深入理解半、次黎曼流形的几何特性,开展此类空间中的诸如Killing场论的进一步研究,以及度规的内蕴“弯曲性”及运动不变性等等,是十分重要和迫切的.
本文拟主要研究半、次黎曼流形上射影和共形映照下的几何不变性及相关的应用问题.
文章的第一部分包括第二章和第三章,第二章讨论了伪对称半黎曼流形上的射影映照的性质.对具有半对称联络的伪对称半黎曼流形,给出了其上关于射影映照的一整体结果：伪对称的半黎曼流形关于射影映照形成了一个闭类;同时给出了伪对称半黎曼流形上的射影等价度量的分类.
第三章研究了半黎曼流形上保持时空模型广义拟Einstein空间不变的一类共形映照,在此类共形映照下,我们得到了一类具有几何特征的空间曲线-椭圆是保持不变的,称之为ξ,η-拟保圆映照,且得出该类映照能够把动力系统中的椭圆动力系变到椭圆动力系.此外,还得到了ξ,,η-拟保圆映照下的不变量,根据此不变量的性质,我们得出ξ,η-拟保圆映照的保持广义拟Einstein空间的不变性；另外,我们给出了容有ξ,η-拟保圆映照的流形具有的几何特征,特别的,研究了一大类容有ξ,η-拟保圆映照的循环流形的几何结构,包括Ruse循环、共圆循环、共调和循环、共形循环空间,得出容有ξ,η拟保圆映照的这一大类循环流形都约化为拟Einstein空间.
文章第二部分即第四章,主要研究次黎曼流形中的次共形映照下的不变性问题以及射影等价的非完整联络的特性.给出了“水平Einstein"(?)口“水平拟Einstein"空间的定义,并刻画了两种空间的几何特征.
对于次黎曼空间中的次共形映照,首先,找到了次共形映照下的几类几何不变量;
其次,研究并给出了次共形循环次黎曼流形约化为次共形平坦的次黎曼流形的充要条件以及次共形映照保持“水平Einstein"空间不变的充要条件；
最后,研究了次黎曼流形上的射影等价的非完整联络,给出了次黎曼流形上两个不同的非完整联络具有相同测地线的充要条件.
本文获得的主要结论如下：对于伪对称半黎曼空间上的射影映照我们有
定理2.2.1假设Ψ：(M,g)→(丽,动是从伪对称半黎曼空间(M,g)到半黎曼空间(M,g)的半对称射影映照,则半黎曼空间(M,g)也是伪对称的.
对于ξ,η-拟保圆映照,我们有
定理3.2.1如果两个半黎曼流形(M,g)和(M,g)成共形对应g=ρ2g,则该共形映照是ξ,η-拟保圆映照的充要条件是张量G或者其缩并张量G是不变的.
同时,ξ,η-拟保圆映照可以保持某些具有几何特征的空间曲线不变.
定理3.3.2如果在半黎曼流形M、M存在共形映照gij=ρ2gij,且函数ρ满足下面方程其中a，b,c是M上的任意函数.则在这类共形映照下,M中每个切方向为Mij的主方向的椭圆都变到M中的椭圆.
对于次黎曼流形中的次共形映照,我们找到了该映照下的几类几何不变量：水平托马斯联络系数、次Weyl共形曲率张量以及次共形第二基本张量,将文中定理4.2.1,4.2.2,4.2.3统一起来可叙述如下
定理在次共形映照下，次黎曼流形的水平托马斯联络系数、次Weyl共形曲率张量、次共形第二基本张量都是不变量.
特别地,由次共形第二基本张量的共形不变性,我们得到如下的
定理4.2.5在次黎曼流形的次共形变换下，全水平脐点子流形变为全水平脐点子流形.
对于次黎曼流形上的两个不同的非完整联络所对应的测地线之间的关系研究,得到了下面的
定理4.3.2假设▽,▽是次黎曼流形M上两个非完整联络,则下面结论是等价的
(1)▽和▽有相同的非完整测地线(不同的参数);
(2)对每一个X∈H,存在λx使得D(X,X)=λxX；
(3)存在唯一的1-形式ω使得S(X,Y)=ω(X)Y+ω(Y)X.
A semi-Riemannian manifold is a smooth manifold associated with a non-degenerate met-ric tensor. It is obvious that the Riemannian manifold is a special case of semi-Riemannian manifolds. A remarkable case of semi-Riemannian manifolds is the so-called Lorentz man-ifold. The general relativity spacetime is an exactly connected time-oriented4-dimensional Lorentz space.1905, Einstein had recognized that the gravitation in spacetime is in correspon-dence with the curvature in a semi-Riemannian space, which promotes the thread applications of semi-Riemannian geometry to the special and the general relativity, for cosmology (red-shift, expanding universe, and big bang) and the gravitation of a single star (perihelion precession, bending of light, and black holes).
A sub-Riemannian manifold, roughly speaking, is a smooth manifold associated with a distribution and a fibre-inner product on it. When the distribution is the whole tangent bundle, then the sub-Riemannian manifold reduces to be a Riemannian manifold. On the one hand, as a natural generalization of Riemannian spaces, the sub-Riemannian manifolds constitute the basic spaces for the geometry and analysis in metric spaces, and also provide a possible common research framework for the study of sub-elliptic operators and Cauchy-Riemannian manifolds; On the other hand, the sub-Riemannian manifolds have extensively been important theory and practical applications in control theory, classical mechanics, gauge field theory and quantum physics.
The increasingly extensive applications of semi-and sub-Riemannian manifolds also have contributed significantly to the development of the geometrical spaces, thus many important results are extended to semi-and sub-Riemannian manifolds, such as submanifold geometry of sub-Riemannian manifolds, normal geodesics, sub-projective mappings, sub-conformal map-pings and so on.
Projective and conformal mappings are all equivalent relations, through them one can con-struct projective and conformal invariants, respectively. And we can study the geometric ob-jects classification and identification by the invariants. For instance, the projective equivalent metrics of constant curvature metrics are also of constant curvature; and the projective equiv- alent metrics of complete Einstein metrics are proportional; a connected Lie group acts on a closed connected Riemannian manifold by transformations that preserve the geodesics (as non-parametric curves), then its acts by isometries or the manifold is covered by the round spheres; the projective equivalence of two metrics is reduced to the existence of BM-structures.
As is well known, the symmetry group of n-dimensional Laplace equation is exactly the conformal group in Euclidean space, using the conformal symmetry, one can extend the har-monic functions to the Riemannian manifolds of conformally flat. Axial symmetric metrics can describe the external geometry of spacetime of some class of uncharged rotating planets and so on. For the case of two-dimensional conformal transformation group, one can comprehend the conformal property via the holomorphic functions, and construct the profound Riemannian mapping theorem in complex analysis.
In differential geometry, the research of the transform theories posed by Klein forms the geometric classification of Riemannian spaces. The spaces of constant curvature are particularly important cases of conformally flat Riemannian manifolds, and the research of the spaces of constant curvature leads to the birth of Einstein manifold. Moreover, we know that an isometric mapping keeps the sectional curvature unchanged, but the inverse is not true. If there have the additional restrictions on the geodesics, then inverse is true. This is the famous Cartan isometric theorem. The global generalization is given by Ambrose and Hicks, i.e. Cartan-Ambrose-Hicks theorem.
As mentioned above, the transform theory plays a decisive and substantial role for the re-veal of the geometric characteristics, physical properties of the spaces. Thus it is necessary to continue to carry out the research in this field. In Riemannian geometry, projective mappings and conformal mappings are both important in differential geometry, and the research of them forms a comprehensive theory system, but in the semi-and sub-Riemannian geometry, there are some fundamental problems open. Such as whether there exists the semi-Weyl conformal curvature tensor related to the conformal transformations? Is it also true that? What is the properties of geodesics on the semi-Riemannian manifolds admitting concircular transforma-tion? Whether the first fundamental form of a surface is proportional under a sub-conformal transformation? And whether the generalized symmetric spaces are invariant under the projec- tive mapping and the generalized Einstein spaces are unchanged under the conformal mappings? Moreover, the conformal invariants and the projective equivalent connection in sub-Riemannian spaces are not resolved completely. These questions are basic, but of significant meanings. Thus it is very important to make clear all the problems mentioned above, which can leads to further understanding of the geometric characteristics of semi-and sub-Riemannian manifolds, and the further research of Killing field, the intrinsic bending, the invariance under the motions and so on.
This paper is concentrated on the invariance study of conformal and projective mappings in semi-and sub-Riemannian manifolds and their applications.
This paper consists of two parts, and is separated into four chapters. The first part contains the second and the third chapters. In the second chapter, we discuss the projective mappings of semi-Riemannian manifolds, especial the pseudo-symmetric semi-Riemannian manifolds admitting a semi-symmetric connection. We obtain a global result:The pseudo-symmetric semi-Riemannian manifolds form a closed class with respect to projective mappings, more-over, we give the classification of the projective equivalent metrics on pseudo-symmetric semi-Riemannian manifolds.
In the third chapter, we investigate a class of conformal mappings of semi-Riemannian manifolds which can keep generalized quasi-Einstein manifolds unchanged. We show that every ellipse in semi-Riemannian manifolds is transformed into a ellipse under this class of confor-mal mappings, so we call this type of conformal mappings the ξ,η-quasi concircular mappings. Moreover, we obtain that this class of conformal mappings take elliptic dynamical systems into elliptic dynamical systems. We also obtain the corresponding invariant under such a mapping, by means of the properties of the invariants, we get an ξ,η-quasi concircular mapping keeps a generalized quasi-Einstein manifold unchanged. Then we characterize the geometrical proper-ties of the semi-Riemannian manifolds admitting ξ,η-quasi concircular mappings, in particular, we study a big class of recurrent manifolds, including Ruse recurrent, concircularly recurrent, conharmonically recurrent, conformally recurrent, we prove that this big class of recurrent man-ifolds admitting ξ,η-quasi concircular mappings are all reduced to quasi-Einstein spaces.
The second part of the paper is exact the fourth chapter, we mainly study the geometrical invariance of sub-conformal mappings and the projective equivalent nonholonomic connections in sub-Riemannian manifolds. We introduce the horizontal Einstein space and the horizontal quasi-Einstein space and explain the geometrical characteristics of the two spaces. We also discuss the relation between the horizontal quasi-Einstein spaces and some symmetric sub-Riemannian manifolds. For the sub-conformal mappings of sub-Riemannian manifolds, we obtain several classes of invariants under the sub-conformal mappings, then we give a necessary and sufficient conditions of a sub-conformally recurrent sub-Riemannian manifold reducing to a sub-conformally symmetric space and a sub-conformal mapping keeping a horizontal Einstein space invariant. In the last, we discuss when two different nonholonomic connections on a sub-Riemannian manifold have the same geodesics, we give the necessary and sufficient conditions of two different nonholonomic connections have the same geodesics.
The main results of this paper are as follows:
For the projective mappings of pseudo-symmetric semi-Riemannian manifolds we have
Theorem2.2.1IfΨ:(M,g)→(M,g) is the semi-symmetric projective mapping from pseudo-symmetric semi-Riemannian manifold (M, g) to a semi-Riemannian manifold (M,g), then (M,g) is also a psendo-symmetric manifold.
For ξ,η-quasi concircular mappings, we get:
Theorem3.2.1Suppose that there exists an conformal mapping g=ρ2g between two semi-Riemannian manifolds (M,g) and (M,g), then, the conformal mapping is ξ, η-quasi concircular if and only if the tensor G or the contracted tensor G are invariants.
Moreover, we find there are the corresponding curves having some geometrical character-istics under this mapping:
Theorem3.3.2If there is a conformal mapping gij=ρ2gij between two semi-Riemannian manifolds M and M, and the function p satisfies the following equations where a, b, c are any scalars on M. Then, under this mapping, every ellipse in M with tangent direction being the principle direction of Mij is transformed to a ellipse in M.
For the sub-conformal mappings in sub-Riemannian manifolds, we find several classes of sub-conformal invariants:horizontal Thomas coefficients of connections, Weyl sub-conformal curvature tensor, the sub-conformal second tensor. From theorem4.2.1,4.2.2,4.2.3we have Theorem Under the sub-conformal mappings, the horizontal Thomas coefficients of connec-tion, the Weyl sub-conformal curvature tensor, the sub-conformal second tensor are all invari-ants.
In particular, in view of the conformal invariance of sub-conformal second tensor, we get
Theorem4.2.4The sub-conformal mappings take horizontal totally umbilical submanifolds to horizontal totally umbilical submanifolds.
At the last of the paper, we study the necessary and sufficient conditions of two different nonholonomic connections having the same geodesics.
Theorem4.3.2Suppose (?)and (?)are two nonholonomic connections on sub-Riemannian mani-fold M, then the following are equivalent
(1)(?) and (?)have the same geodesics (with the different parameters);
(2)for every X∈H, there exists λx such that D(X, X)=λXX;
(3)there exist a unique1-form ω such that S(X,Y)=ω(X)Y+ω(Y)X.
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