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Riemannian Geometry of Contact and Symplectic Manifolds
08 Jan 2002-
TL;DR: In this article, the authors describe a complex geometry model of Symplectic Manifolds with principal S1-bundles and Tangent Sphere Bundles, as well as a negative Xi-sectional Curvature.
Abstract: Preface * 1. Symplectic Manifolds * 2. Principal S1-bundles * 3. Contact Manifolds * 4. Associated Metrics * 5. Integral Submanifolds and Contact Transformations * 6. Sasakian and Cosymplectic Manifolds * 7. Curvature of Contact Metric Manifolds * 8. Submanifolds of Kahler and Sasakian Manifolds * 9. Tangent Bundles and Tangent Sphere Bundles * 10. Curvature Functionals and Spaces of Associated Metrics * 11. Negative Xi-sectional Curvature * 12. Complex Contact Manifolds * 13. Additional Topics in Complex Geometry * 14. 3-Sasakian Manifolds * Bibliography * Subject Index * Author Index
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15 Apr 2021
TL;DR: In this article, it was shown that Bach flat almost coKahler manifold admits Ricci solitons, satisfying the critical point equation (CPE) or Bach flat.
Abstract: In this paper, we study an almost coKahler manifold admitting certain metrics such as $$*$$
-Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKahler 3-manifold (M, g) admitting a $$*$$
-Ricci soliton (g, X) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKahler $$(\kappa ,\mu )$$
-almost coKahler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f. Finally, we prove that a $$(\kappa , \mu )$$
-almost coKahler manifold (M, g) is coKahler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKahler manifolds which are non-coKahler.
6 citations
TL;DR: In this paper, the authors obtained different characterizations of the spherical strictly pseudoconvex CR manifolds admitting a CR-symmetric Webster metric by means of the Tanaka-Webster connection and of the Riemannian curvature tensor.
Abstract: In this paper we get different characterizations of the spherical strictly pseudoconvex CR manifolds admitting a CR-symmetric Webster metric by means of the Tanaka–Webster connection and of the Riemannian curvature tensor. As a consequence we obtain the classification of the simply connected, spherical symmetric pseudo-Hermitian manifolds.
6 citations
Posted Content•
TL;DR: In this paper, the authors considered the problem of deformation of curves of fixed degree and gave a sufficient condition to guarantee the possibility of deforming a curve, which is equivalent to the surjectivity of a holonomy map.
Abstract: We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and this condition is expressed via a differential equation along the curve. In the classical differential geometry setting, the analogous condition was considered by Bryant and Hsu in [Invent. Math., 114(2):435-461, 1993, J. Differential Geom., 36(3):551-589, 1992], who proved that it is equivalent to the surjectivity of a holonomy map. The purpose of this paper is to extend this deformation theory to curves of fixed degree providing several examples and applications. In particular, we give a useful sufficient condition to guarantee the possibility of deforming a curve.
6 citations
Cites background from "Riemannian Geometry of Contact and ..."
...The structure given by (M,ω, g, J) is called a contact Riemannian manifold, see [5] and [14]....
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05 Sep 2007
TL;DR: In this article, a new stratification of the contact quotient at zero, called the C-L stratification, is introduced, which is a morphism of stratified spaces.
Abstract: This thesis deals with applications of Lie symmetries in differential geometry and dynamical systems. The first chapter of the thesis studies the singular reduction of symmetries of cosphere bundles, the conservation properties of contact systems and their reduction. We generalise the results of [15] to the singular case making a complete topological and geometrical analysis of the reduced space. Applying the general theory of contact reduction developed by Lerman and Willett in [33] and [57], one obtains contact stratified spaces that lose all information of the internal structure of the cosphere bundle. Based on the cotangent bundle reduction theorems, both in the regular and singular case, as well as regular cosphere bundle reduction, one expects additional bundle-like structure for the contact strata. The cosphere bundle projection to the base manifold descends to a continuous surjective map from the reduced space at zero to the orbit quotient of the configuration space, but it fails to be a morphism of stratified spaces if we endow the reduced space with its contact stratification and the base space with the customary orbit type stratification defined by the Lie group action. In this chapter we introduce a new stratification of the contact quotient at zero, called the C-L stratification (standing for the coisotropic or Legendrian nature of its pieces) which solves the above mentioned two problems. Its main features are the following. First, it is compatible with the contact stratification of the quotient and the orbit type stratification of the configuration orbit space. It is also finer than the contact stratification. Second, the natural projection of the C-L stratified quotient space to its base space, stratified by orbit types, is a morphism of stratified spaces. Third, each C-L stratum is a bundle over an orbit type stratum of the base and it can be seen as a union of C-L pieces, one of them being open and dense in its corresponding contact stratum and contactomorphic to a cosphere bundle. The other strata are coisotropic or Legendrian submanifolds in the contact components that contain them. We also describe the relation between contact vector fields and the time dependent Hamilton-Jacobi equation. The reduction of contact systems and time dependent Hamiltonians is mentioned. In the second chapter we study geometric properties of Sasakian and Kahler quotients. We construct a reduction procedure for symplectic and Kahler manifolds using the ray preimages of the momentum map. More precisely, instead of taking as in point reduction the preimage of a momentum value μ, we take the preimage of ℝ+μ, the positive ray of μ. We have two reasons to develop this construction. One is geometric: non zero Kahler point reduction is not always well defined. The problem is that the complex structure may not leave invariant the horizontal distribution of the Riemannian submersion πμ : J-1(μ) → Mμ. The solution proposed in the literature is correct only in the case of totally isotropic momentum (i.e. Gμ = G). The other reason is that it provides invariant submanifolds for conformal Hamiltonian systems. They are usually non-autonomous mechanical systems with friction whose integral curves preserve, in the case of symmetries, the ray pre-images of the momentum map. We extend the class of conformal Hamiltonian systems already studied and complete the existing Lie Poisson reduction with the general ray one. As examples of symplectic (Kahler) and contact (Sasakian) ray reductions we treat the case of cotangent and cosphere bundles and we show that they are universal for ray reductions. Using techniques of A. Futaki, we prove that, under appropriate hypothesis, ray quotients of Kahler-Einstein or Sasaki-Einstein manifolds remain Kahler or Sasaki-Einstein. Note that it suffices to prove the Kahler case and the compatibility of ray reduction with the Boothby-Wang fibration. In the last chapter, we prove a stratification theorem for proper groupoids. First we find an equivalent way of describing the same result for a proper Lie group action, way which uses the theory of foliations and can be adapted to the language of Lie groupoids. We treat separately the case of free and proper groupoids. The orbit foliation of a proper Lie groupoid is a singular Riemannian foliation and we show this explicitly.
6 citations
Posted Content•
TL;DR: In this paper, the authors define a left invariant Riemannian metric on a nilpotent Lie group to be minimal if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar cur- vature.
Abstract: Let (N,) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their 'almost' versions). We define a left invariant Riemannian metric on N com- patible with to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar cur- vature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the 'invariant Ricci' flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two geometric structures with Riemannian data, giving rise to a great deal of invariants. Our approach proposes to vary Lie brackets rather than inner products; our tool is the moment map for the action of a reductive Lie group on the algebraic variety of all Lie algebras, which we show to coincide in this setting with the Ricci operator. This gives us the possibility to use strong results from geomet- ric invariant theory. We describe the moduli space of all isomorphism classes of geometric structures on nilpotent Lie groups of a given class and dimension admitting a minimal compatible metric, as the disjoint union of semi-algebraic varieties which are homeomorphic to categorical quotients of suitable linear ac- tions of reductive Lie groups. Such special geometric structures can therefore be distinguished by using invariant polynomials.
6 citations