Smooth Banach structure on orbit spaces and leaf spaces

Motivated by the desire to quantize singular symplectic spaces we consider stratified spaces from an analytic and geometric point of view. To this end one needs an appropriate functional structure on these spaces. But unlike for manifolds such a functional structure on a stratified space is in general not intrinsically given. In this article we explain the basic notions of the theory of stratified spaces and define an appropriate concept for a so-called smooth (functional) structure on a stratified space. We explain how varieties, orbit spaces and reduced spaces of Hamiltonian group actions give rise to natural examples for stratified spaces with a smooth structure. Moreover, it is shown how smooth structures allow for the definition of geometric concepts on stratified spaces like tangent spaces, vector fields and Poisson bivectors. Finally, it is explained what to understand by the quantization of a symplectic stratified space.KeywordsOrbit SpaceOrbit TypeSmooth StructureSmooth Vector FieldPoisson BivectorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this work, by a dynamical system we mean a pair $(S, \,X)$, where $S$ is either a pseudogroup of local diffeomorphisms, or a transformation group, or a smooth foliation of the manifold $X$. The groups of transformations can be both discrete and nondiscrete. We define the concepts of attractor and global attractor of the dynamical system $(S, \,X)$ and investigate the properties of attractors and the problem of the existence of attractors of dynamical systems $(S, \,X)$. Compactness of attractors and ambient manifolds is not assumed. A property of the dynamical system is called transverse if it can be expressed in terms of the orbit space or the leaf space (in the case of foliations). It is shown that the existence of an attractor of a dynamical system is a transverse property. This property is applied by us in proving two subsequent criteria for the existence of an attractor (and global attractor): for foliations of codimension $q$ on an $n$-dimensional manifold, $0 < q < n$, and for foliations covered by fibrations. A criterion for the existence of an attractor that is a minimal set for an arbitrary dynamical system is also proven. Various examples of both regular attractors and attractors of transformation groups that are fractals are constructed.

In this paper we prove that an isometry between orbit spaces of two proper isometric actions is smooth if it preserves the codimension of the orbits or if the orbit spaces have no boundary. In other words, we generalize Myers–Steenrod’s theorem for orbit spaces. These results are proved in the more general context of singular Riemannian foliations.

We study some cases when the sectional curvature remains positive under the taking of quotients by certain nonfree isometric actions of Lie groups. We consider the actions of the groups $S^1$ and $S^3$ such that the quotient space can be endowed with a smooth structure using the fibrations $S^3/S^1{\simeq}S^2$ and $S^7/S^3\simeq S^4$. We prove that the quotient space carries a metric of positive sectional curvature, provided that the original metric has positive sectional curvature on all 2-planes orthogonal to the orbits of the action.

We prove that the boundary of an orbit space or more generally a leaf space of a singular Riemannian foliation is an Alexandrov space in its intrinsic metric, and that its lower curvature bound is that of the leaf space. A rigidity theorem for positively curved leaf spaces with maximal boundary volume is also established and plays a key role in the proof of the boundary problem.

Let $F$ be a leafwise hyperbolic taut foliation of a closed 3-manifold $M$ and let $L$ be the leaf space of the pullback of $F$ to the universal cover of $M$. We show that if $F$ has branching, then the natural action of $\pi_1(M)$ on $L$ is faithful. We also show that if $F$ has a finite branch locus $B$ whose stabilizer acts on $B$ nontrivially, then the stabilizer is an infinite cyclic group generated by an indivisible element of $\pi_1(M)$.

Pursell and Shanks [8] proved that a Lie algebra isomorphism between Lie algebras of all C°° vector fields with compact support on paracompact connected C°° manifolds M and N yields a diffeomorphism between the manifolds M and N. Similar results hold for some other structures on manifolds. Indeed, Omori [6] proved the corresponding results in the case of volume structures, symplectic structures, contact structures and fibering structures with compact fibers. The case of complex structures was studied by Amemiya [1]. Koriyama [5] proved that in the case of Lie algebras of vector fields with invariant submanifolds. Recently, Fukui [4] studies the case of Lie algebras of G-invariant C°° vector fields with compact support on paracompact free smooth Gmanifolds when G is a compact connected semi-simple Lie group. The corresponding result is no longer true when G is not semi-simple or G does not act freely. In this paper, we consider Pursell-Shanks type theorem for orbit spaces of smooth G-manifolds in the case of G a compact Lie group. For a smooth G-manifold M, the orbit space M/G inherits a smooth structure by defining a function on M/G to be smooth if it pulls back to a smooth function on M, and the Zariski tangent space of M/G can be defined. This smooth structure of the orbit space was studied by Schwarz [9], [11], Bierstone [2], Poenaru [7] and Davis [3], Schwarz [10] defined a Lie algebra 9£ (M/G) of smooth vector fields on the orbit

Smooth functions invariant under the action of a compact lie group

As recounted in this paper, the idea of groups is one that has evolved from some very intuitive concepts. We can do binary operations like adding or multiplying two elements and also binary operations like taking the square root of an element (in this case the result is not always in the set). In this paper, we aim to find the operations and actions of Lie groups on manifolds. These actions can be applied to the matrix group and Bi-invariant forms of Lie groups and to generalize the eigenvalues and eigenfunctions of differential operators on Rn. A Lie group is a group as well as differentiable manifold, with the property that the group operations are compatible with the smooth structure on which group manipulations, product and inverse, are distinct. It plays an extremely important role in the theory of fiber bundles and also finds vast applications in physics. It represents the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. Here we did work flat out to represent the mathematical aspects of Lie groups on manifolds.

The standard Lichnerowicz comparison theorem states that if the Ricci curvature of a closed, Riemannian n-manifold M satisfies Ric (X, X) > a(n - 1) |X| 2 for every X E TM for some fixed a > 0, then the smallest positive eigenvalue A of the Laplacian satisfies A > an. The Obata theorem states that equality occurs if and only if M is isometric to the standard n-sphere of constant sectional curvature a. In this paper, we prove that if M is a closed Riemannian manifold with a Riemannian foliation of codimension q, and if the normal Ricci curvature satisfies Ric⊥ (X,X) ≥ a (q - 1) |X| 2 for every X in the normal bundle for some fixed a > 0, then the smallest eigenvalue λ B of the basic Laplacian satisfies λ B > aq. In addition, if equality occurs, then the leaf space is isometric to the space of orbits of a discrete subgroup of O (q) acting on the standard q-sphere of constant sectional curvature a. We also prove a result about bundle-like metrics on foliations: On any Riemannian foliation with bundle-like metric, there exists a bundle-like metric for which the mean curvature is basic and the basic Laplacian for the new metric is the same as that of the original metric.

This overview is intended as a lightweight companion to the long article [20]. One of the main results there is the determination of the rational cohomology of the stable mapping class group, in agreement with the Mumford conjecture [26]. This is part of a recent development in surface theory which was set in motion by Ulrike Tillmann’s discovery [34] that Quillen’s plus construction turns the classifying space of the stable mapping class group into an infinite loop space. Tillmann’s discovery depends heavily on Harer’s homological stability theorem [15] for mapping class groups, which can be regarded as one of the high points of geometric surface theory. 1. Surface bundles without stable homotopy theory We denote by Fg,b an oriented smooth compact surface of genus g with b boundary components; if b = 0, we also write Fg. Let Diff(Fg,b; ∂) be the topological group of all diffeomorphisms Fg,b → Fg,b which respect the orientation and restrict to the identity on the boundary. (This is equipped with the Whitney C∞ topology.) Let Diff1(Fg,b; ∂) be the open subgroup consisting of those diffeomorphisms Fg,b → Fg,b which are homotopic to the identity relative to the boundary. Theorem 1.1. [10], [11]. If g > 1 or b > 0, then Diff1(Fg,b; ∂) is contractible. Idea of proof. For simplicity suppose that b = 0, hence g > 1. Write F = Fg = Fg,0. Let H(F ) be the space of hyperbolic metrics (i.e., Riemannian metrics of constant sectional curvature −1) on F . The group Diff1(F ; ∂) acts on H(F ) by transport of metrics. The action is free and the orbit space is the Teichmuller space T(F ). The projection map H(F ) −→ T(F ) admits local sections, so that H(F ) is the total space of a principal bundle with structure group Diff1(F ; ∂). By Teichmuller theory, T(F ) is homeomorphic to a euclidean space, hence contractible. It is therefore enough to show that H(F ) is contractible. This is not easy. Let S(F ) be the set of conformal structures on F (equivalently, complex manifold structures on F which refine the given smooth structure and are compatible with the orientation of F ). Let J(F ) be the set of almost complex structures on F . Elements of J(F ) can be regarded as smooth vector bundle automorphisms J : TF → TF with the property J = −id and det(a(v), aJ(v)) > 0 for any x ∈ F , v ∈ TxF and oriented isomorphism a :TxF → R. Hence J(F ) has

This is the second in a series of papers devoted to GM-manifolds and their unique structure. We consider the GM-set associated with a given transformation group G acting on the smooth manifold M. The primary goal is detailed analysis of GM-sets in connection with the underlying transformation groups and providing a rigorous theoretical justification of GM-sets. Then, we will investigate the existence of unique manifold topology and smooth structure on each GM-set such that the induced-action of the Lie group G on GM-set is smooth.

Lorentzian manifolds with shearfree congruences and Kähler-Sasaki geometry

Let X X be a PL homotopy C P 2 k + 1 C{P^{2k + 1}} corresponding by Sullivan’s classification to the element ( N 1 , α 2 , N 2 , ⋯ , α k , N k ) ({N_1},{\alpha _2},{N_2}, \cdots ,{\alpha _k},{N_k}) of Z ⊕ Z 2 ⊕ Z ⊕ ⋯ ⊕ Z 2 ⊕ Z Z \oplus {Z_2} \oplus Z \oplus \cdots \oplus {Z_2} \oplus Z . Theorem 1. The topological circle action on S 4 k + 3 {S^{4k + 3}} with orbit space X X is the restriction of an S 3 {S^3} action with a triangulable orbit space iff α i = 0 , i = 2 , ⋯ , k {\alpha _i} = 0,i = 2, \cdots ,k ; and N 1 ≡ 0 mod 2 {N_1} \equiv 0\bmod 2 ; and ∑ ( − 1 ) i N i = 0 \sum {( - 1)^i}{N_i} = 0 . If X X admits a smooth structure and satisfies the hypotheses of Theorem 1, a certain smoothing obstruction arising from the integrality theorems vanishes for the corresponding S 3 {S^3} action.

The Hamiltonian actions of $\S ^{1}$ on the symplectic manifold $\mathbb {R}^{6}$ in the $1:1:-2$ and $1:1:2$ resonances are studied. Associated to each action is a Hilbert basis of polynomials defining an embedding of the orbit space into a Euclidean space $V$ and of the reduced orbit space $J^{-1}(0)/\S ^{1}$ into a hyperplane $V_{J}$ of $V$, where $J$ is the quadratic momentum map for the action. The orbit space and the reduced orbit space are singular Poisson spaces with smooth structures determined by the invariant functions. It is shown that the Poisson structure on the orbit space, for both the $1:1:2$ and the $1:1:-2$ resonance, cannot be extended to $V$, and that the Poisson structure on the reduced orbit space $J^{-1}(0)/\S ^{1}$ for the $1:1:-2$ resonance cannot be extended to the hyperplane $V_{J}$.