# Some Applications of Symmetries in Differential Geometry and Dynamical Systems

Abstract : My research lies at the interface of Riemannian, contact, and symplectic geometry. It deals with the construction of Kähler and Sasaki-Einstein metrics, with the study of conformal Hamiltonian systems, the geometry of cosphere bundles, and proper Lie groupoids. The main theme of this thesis is the study of applications of Lie symmetries in differential geometry and dynamical systems. The first chapter of the thesis studies the singular reduction of cosphere fiber bundles. The copshere bundle of a differentiable manifold $M$ (denoted by $S^*(M)$) is the quotient of its cotangent bundle without the zero section with respect to the action by multiplications of $\RR^+$ which covers the identity on $M$. It is a contact manifold which has the same privileged position in contact geometry that cotangent bundles have in symplectic geometry. Using a Riemannian metric on $M$, we can identify $S^*(M)$ with its unitary tangent bundle and its Reeb vector field with the geodesic field on $M$. If $M$ is endowed with the proper action of a Lie group $G$, the lift of this action on $S^*(M)$ respects the contact structure and admits an equivariant momentum map $J$. We study the topological and geometrical properties of the reduced space of $S^*(M)$ at zero momentum, i.e. $\left(S^*(M)\right)_0 :=J^{-1}(0)/G$. Thus, we generalize the results of \cite{dragulete--ornea--ratiu} to the singular case. Applying the general theory of contact reduction developed by Lerman and Willett in \cite{lerman--willett} and \cite{willett}, one obtains contact stratified spaces that lose all information of the internal structure of the cosphere bundle. Even more, the cosphere bundle projection to the base manifold descends to a continuous surjective map from $\left(S^*(M)\right)_0$ to $M/G$, but it fails to be a morphism of stratified spaces if we endow $\left(S^*(M)\right)_0$ with its contact stratification and $M/G$ with the customary orbit type stratification defined by the Lie group action. 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. To solve these problems, we introduce a new stratification of the contact quotient at zero, called the \textit{C-L stratification} (standing for the coisotropic or Legendrian nature of its pieces). It is compatible with the contact stratification of $\left(S^*(M)\right)_0$ and the orbit type stratification of $M/G$. It is also finer than the contact stratification. Also, the natural projection of the C-L stratified quotient space $\left(S^*(M)\right)_0$ to its base space, stratified by orbit types, is a morphism of stratified spaces. 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. Hence we have identified the maximal strata endowed with cosphere bundle structure. The other strata are coisotropic or Legendrian submanifolds in the contact components that contain them. Consequently, we can perform a complete geometric and topological analysis of the reduced space. We also study the behaviour of the projection on $\left(S^*(M)\right)_0$ of the Reeb flow (geodesic flow). The set of contact Hamiltonian vector fields (the analogous of Hamiltonian vector fields in symplectic geometry) form the "Lie" group of the algebra of contact transformations. In the first chapter we also present the reduction of contact systems (which locally are in bijective correspondence with the non-autonomus Hamilton-Jacobi equations) and time dependent Hamiltonian systems. In the second chapter of this thesis we study quotients of Kähler and Sasaki-Einstein manifolds. We construct a reduction procedure for symplectic and Kähler manifolds (endowed with symmetries generated by a Lie group) which uses the ray pre-images of the associated momentum map. More precisely, instead of considering as in the Marsden- Weinstein reduction (point reduction) the pre-image of a momentum value $\mu$, we use the pre-image of $\RR^+\mu$, its positive ray. We have three reasons to develop this construction. One is geometric: the construction of canonical reduced spaces of Kähler manifolds corresponding to a non zero momentum. By canonical we mean that the reduced Kähler structure is the projection of the initial Kähler structure. The point reduction (Marsden-Weinstein) given by $M_\mu:=\frac{J^{-1}(\mu)}{G_\mu}$, where $\mu$ is a value of the momentum map $J$ and $G_\mu$ the isotropy subgroup of $\mu$ with respect to the coadjoint action of $G$ is not always well defined in the Kähler case (if $G\neq G_\mu$). The problem is caused by the fact that the complex structure of $M$ does not leave invariant the horizontal distribution of the Riemannian submersion which projects $J^{-1}(\mu)$ on $M_\mu$. The solution proposed in the literature uses the reduced space at zero momentum of the symplectic difference of $M$ with the coadjoint orbit of $\mu$ endowed with a unique Kähler-Einstein form (constructed, for insatnce, in \cite{besse}, Chapter $8$) and different from the Kostant-Kirillov-Souriau form. The uniqueness of the form on the coadjoint orbit ensures that the reduced space is well defined. On the other hand, not using the Kostant-Kirillov-Souriau form implies the fact that the reduced space is no longer canonical. The ray reduced space that we construct is canonical and can be defined for any momentum. It is the quotient of $J^{-1}(\RR^+\mu)$ with respect to a certain normal subgroup of $G_\mu$. The second reason is an application to the study of conformal Hamiltonian systems (see \cite{mclachlan--perlmutter}). They are mechanical, non-autonomous systems with friction whose integral curves preserve, in the case of symmetries, the ray pre-images of the momentum map, but not the point (momentum) preimages of the Marsden-Weinstein quotient. We extend the notion of conformal Hamiltonian vector field by showing that one can thus include in this study new mechanical systems. Also, we present the reduction of conformal Hamiltonian systems. The third reason consists of finding the necessary and sufficient conditions for the ray reduced spaces of Kähler (Sasakian)-Einstein manifolds to be also Kähler (Sasakian)-Einstein. We deal with this problem in the second chapter of the thesis, in \cite{dragulete--ornea}, and in \cite{dragulete--doi} where we use techniques of A. Futaki. Thus, we can construct new Sasaki-Einstein structures. As examples of symplectic (Kähler) and contact (Sasakian) ray quotients we treat the case of cotangent and cosphere bundles and show that they are universal spaces for ray reductions. Examples of toric actions on spheres are also described. The third chapter of my thesis studies the space of orbits of a proper Lie groupoid. In \cite{weinstein--unu}, \cite{weinstein--doi} A. Weinstein has partially solved the problem of linearization of proper groupoids. In \cite{zung}, N. T. Zung has completed it by showing a theorem of Bochner type for proper groupoids. Using ideas from foliation theory and the slice (linearization) theorem of Weinstein and Zung, we prove a stratification theorem for the orbit space of a proper groupoid. We show explicitely that the orbital foliation of a proper Lie groupoid is a Riemannian singular foliation in the sense of Molino. For all these we have two motivations. On one hand we want to prove that there is an equivalence between proper groupoids and orbispaces (the spaces which are locally quotients with respect to an action of a compact Lie group). On the other hand we would like to study the reduction of infinitesimal actions (actions of Lie algebras) which are not integrable to Lie group actions. These actions and their integrability have been studied, among others, by Palais (\cite{palais}), Michor, Alekseevsky.
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Submitted on : Wednesday, April 23, 2008 - 11:55:08 PM
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### Citation

Oana Dragulete. Some Applications of Symmetries in Differential Geometry and Dynamical Systems. Mathematics [math]. Ecole Polytechnique federale de Lausanne, 2007. English. ⟨tel-00275462⟩

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