On The Invariance Of Conformal And Projective Mappings In Semi-and Sub-Riemannian Manifolds

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 haveTheorem2.2.1IfΨ:(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 getTheorem4.2.4The 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.