Last updated 10/March/2012: This article is now finished. See

http://zung.zetamu.net/2012/03/geometry-of-rn-actions-on-n-manifolds-2012/

19/02/2012: We’re behind schedule now. Need to speed up !!!

This is the sketch of a research article in progress with a student of mine. I’m making the sketch of what we want to put in the article, and he takes care of writing and filling in the details. It will be modified many times, until the article is done.

Starting date: 17/Jan/2012

Expected finishing date: before 29/Feb/2012

(we already have some original results to show in the article. In the writing process we’ll try to obtain some additional results)

—–

**1) Introduction**

Motivations

Main results

The results are not very difficult, and some of the results presented here have been known before. But we believe that they are useful, and we want to study these systems in a systematic way, as part of a larger program on the topology of integrable non-Hamiltonian systems.

Structure of the paper

**2) Nondegenerate singularities**

2.1. Definition of nondegenerate singularities.

*2.2. Why nondegeneracy conditions ?*

Our nondegeneracy condition is consistant with the general case of integrable (Hamiltonian or non-Hamiltonian) dynamical systems

Equivalent to Camacho-Chaperon hyperbolic condition in the case of action of Rn on n-manifolds ?

If no nondegeneracy condition then the action can be very degenerate:

**Theorem**: for any open set U in Rn there exist n commuting vector fields on Rn which are independent almost everywhere in U, and equal to zero outside U.

This theorem may be “folkloric” and I don’t know who first proved it, but at least one can find it in a paper by Alan Weinstein (about Poisson geometry of real simple Lie algrebas)

Proof: Fill (a dense subset of U) by a union of disjoint balls –> solve the problem for each ball. Fix a diffeo F from Rn to ball , which is “very flat at infinity” (do it in a SO(n) equivariant way –> a function from R+ to bounded interval; function given by an intergal formula, with the integrand a function which decreases very fast at infinity). Project the constant vector fields on Rn to the ball by this map F –> get a set of commuting vector fields on the ball which are flat at the boundary of the ball.

Compare to (CT) condition of Sabatini in the case n=2:

(CT) condition of Sabatini

2 commuing v.f. X, Y in dim 2 such that:

1) have only isolated singular points

2) X(p) = 0 if and only if Y(p) = 0

3) X and Y are independent at every non-zero point (i.e. no rank-1 point)

note that (CT) condition is very different from the condition that toric degree >=1

in fact, there exists system which satisfy (CT) condition, but which doesn’t admit a T1 action. example of S2 with just 2 orbits: a point and the rest (see Case 1 in the thm 3.3 of Sabatini 1997)

but:

if CT and nondegenerate then toric degree >=1

Compare to Morse-Smale condition:

Camacho defines Morse-Smale condition for R2 actions as follows:

1) the set of nonwandering points of the action is an embedded cell complex consisting of finitely many singular orbits

2) The compact orbits are hyperbolic

Camacho shows that Morse-Smale R2-actions exist on any compact orientable 2-manifold

In fact, Camacho’s Morse-Smale systems are the same as our nondegenerate systems.

*2.3. Local normal form*

**Theorem** (unique local normal form). There exist a basis X1, …, Xn of the infinitesimal action, a local coordinate system (x1,…xn), and nonnegative integers h, e, r with (which depend only on the action) such that h+2e + r = n and:

Xi = … (hyperbolic)

Xh+2i-1 = … & Xh+2i = … (elliptic)

Xj = … (regular)

Proof:

1) Invoke Z & Chaperon –> linearization

2) Use classification theorem for linear systems

**Definition.** The pair of numbers (h,e) in the above theorem is called the **HE-invariant** of the singular point.

**Definition.** A basis (X1,…,Xn) of our Rn-action in the above theorem is called a **canonical basis of the action** at the point.

The local canonical basis is not unique. The set of all such bases for the action at a point is given by the following theorem.

**Theorem.** (Canonical bases). Let (X1,…,X_n) be a canonical basis of the action at the point p. Then a basis (Y1,…,Yn) of the Rn-action is also canonical if and only if it satisfies the following conditions:

1) Y1,…,Yh is a permutation of X1,…,Xh

2) The family of couples ((Yh+2i-1,Yh+2i),i=1,…,e) is also a permutation of ((Xh+2i-1,Xh+2i),i=1,…,e), up to possible changes of sign of the Xh+2i-1 (i.e. one can replace Xh+2i-1 by -Xh+2i-1)

(The regular part Yj, j > h+2e is arbitray, there is no condition on it).

Proof. a) Illustrate this theorem in low-dimensional cases

b) General case is the same.

*2.4. The local isometry group of a singular point.*

Theorem. Nondegenerate singular point p of HE-invariant (h,e) and rank r (n= h + 2e + r). Then the group of germs of local isomorphisms (= local diffeomorphisms which preserve the actions) which preserve p is isomorphic to T^e x R^{e+h} x (Z2)^h. The part T^e x R^{e+h} comes from the action \rho itself (internal isometries), and the part (Z2)^h is the “external isometry group”.

*2.5. Nondegenerate singular orbits*

Definition. HERT-invariant (h,e,r,t) of an orbit or a singular point on it: h = number of transversal hyperbolic components,e = number of transversal elliptic components, R^r X T^t = diffeo type of orbit.

An orbit is compact if and only if r = 0, in which case it is a torus of dimension t.

**Theorem.** (Semi-local normal form for compact orbits) Let Op be a compact orbit of HERT-invariant (h,e,0,t). Then a neighborhood of Op will be isomorphic to a neighborhood of 0 in R^h X (D^2)^e X T^t / (Z2)^k for some k <= h with a standard linear action on it (give the precise description of the action here).

Proof:

1) Existence of locally free T^t-action

2) Use local normal form theorem at a point on the orbit –> canonical basis & local coordinates in local normal form. Use theorem about canonical bases –> can assume that the regular part consists of elements which generate the locally free T^t-action

3) Extend the coordinates via this T^k-action from the neighborhood of p toa tubular neighborhood of the compact orbit.

4) Look at the isotropy

Remark: The above theorem is analogous to Miranda-Z theorem on singular orbits of integrable Hamiltonian systems (Miranda-Z 2004)

Theorem. (What orbits can lie the the closure of what orbits ?)

1) If an orbit Op has HERT-invariant (h,e,r,t) with t=0, then any orbit Oq in its closure also has t= 0 in its HERT-invariant

2) …

2.6 The set of singular points

**3) Toric degree**

*3.1. Definition of toric degree*

Definition: Toric degree of an action \rho of Rn is the maximal number k such that the action of Rn descends to an action of Tk x R(n-k)

Another equivalent definiton: toric degree = rank Z_\rho, where Z_\rho is the isotropy group of \rho

Denote the Tk sub-action of \rho by \rho_T

The toric degree can be any number from 0 to n. It is an important invariant of the action. When the toric degree is 0, we will say that the action is *totally hyperbolic*. Totally hyperbolic actions will be studied in the next section. In this section we will show that the toric degree can be read off the HERT invariant of any point, and look at the two special cases of toric degree n and (n-1).

*3.2. Determination of toric degree*

Theorem: Let p be an arbitrary point of M, and denote the HERT invariant of p with respect to the action by (h,e,r,t). Then the toric degree of the action on M is equal to e+t.

*3.3. The case of toric degree n*

Theorem: The only connected closed n-dimensional manifold which admits an effective Rn action of toric degree n is Tn, and the action is unique up to automorphisms of Rn.

*3.4 The case of toric degree (n-1)*

Semi-local picture:

Consider an orbit Op. Then its HERT invariant (h,e,r,t) satisifes:

e+t = n-1

h+2e+r+t =n

The above two equalities imply that h+e+r=1, so there are only 3 possibilities

1) h=e=0,r=1, t=n-1: the orbit is regular = T^{n-1} \times R.

2) h=r=0, e= 1, t=n-2: the orbit is singular = T^{n-2}, tranversally elliptic

3) e=r=0, h=1, t = n-1: the orbit is singular = T^{n-1} transversally hyperbolic

According to ???, Case 3 has two subcases:

3a) the action \rho_T on the singular orbit is free

3b) the action \rho_T on the singular orbit has isotropy = Z2.

Global picture:

(M is assumed to be compact connected)

Look at the orbit space B of the action \rho_T . It is compact, connected, has dimension 1, and can have a finite number of special points on it which correspond to singular orbits of \rho.

Case 2) –> locally B is a half-closed interval which contains the point at the closed end.

Case 3a) –> locally B is an open interval which contains the point inside.

Case 3b) –> locally B is a half-closed interval which contains the point at the closed end.

Thus, topologically, B is either an interval or a circle.

Case B = circle: All singular orbits are of type 3a).The number of singular orbits is strictly positive and even.

Theorem: (Smooth classification in the case B = circle). Complete set of invariants: …

Case B = interval: 2 “special” singular orbits, each of them is of type 2) or 3b), all the other singular orbits are of type 3a). The number of singular orbits of type 3a) is arbitrary (>= 0)

Theorem: (Smooth classification in the case B = interval). Complete set of invariants: …

**4) Actions with hyperbolic singularities**

*4.1. Reflection principle.*

We have the following reflection principle for hyperbolic singularities, which is analogous to the Schwarz reflection principle in complex analysis:

**Theorem**: Let p be a singular point of corank 1 hyperbolic. Then

1) there is an element g of Rn such that the corresponding vector field X = Xg vanishes at p and can be locally written as X = x d/dx near p.

2) p belong to a smooth embedded hypersurface H such that X = 0 on H, and locally near every point of H the vector field X has the type X = x d/dx

3) There is a unique (non-trivial) involution in a neighborhood of H, which preserves the action, and such that H is the set of fixed points of the involution.

*4.2. The closure of a hyperbolic domain.*

Hyperbolic domain = n-dimensional orbit of a totally hyperbolic action.

To describe hyperbolic domains, we will introduce the following notion of H-polytopes.

**Definition**: H-polytope = convex polytope in Rn such that each vertex is locally isomorphic to the standard positive corner of Rn.

Remark: a face of a H-polytope is again a H-polytope. In dimension 2 any polygone is an H-polytope. Starting from dimension 3 the condition for a polytope to be an H-polytope is non-trivial.

Example: A Delzant polytope (in the theory of Hamiltonian toric manifolds) is a H-polytope. But a H-polytope is not a Delzant polytope in general.

Each polytope in Rn is given by a set of inequalities:

where x denotes a point in the polyope, are elements of the dual Rn, and are numbers. We will assume that each fae of the polytope corresponds to exactly one (i.e. there is no redundant inequality).

Question: under which conditions on the polytope will be an H-polytope ?

First condition: are the vertices of a polytope in Rn whose interior contains 0.

Equality: , where is the area of the corresponding hypersurface.

Second condition: b_i are in generic position and satisfy some inequalities ?

**Theorem**: The closure of a hyperbolic domain is diffeomorphic to a H-polytope. Conversely, any H-polytope can be realized.

**Proof**. ???

Invariants of such a closed domain

*4.3 Existence of totally hyperbolic actions.*

Therorem: Any closed smooth n-dimensional manifold admits a completely hyperbolic nondegenerate action of Rn.

5) Low dimensions

This section is devoted to the cases with

*5.1. Two-dimensional case*

When n=2, the toric degree can be 0,1 or 2.

The case toric degree 2 is covered by Theorem ??? In this case the manifold is T2

The case of toric degree 1 is covered by Them ??? In this case the manifold can be either T2 (all singular orbits are T1 non-twisted), S2 (with 2 singular orbits which are elliptic elliptic points, the other singular orbits are hyperbolic T1 non-twisted), Klein bottle (two twisted singular orbits T1, the others are T1 non-twisted), or RP2 (1 fixed point + 1 twisted T1; other singular orbits are T1 non-twisted)

Nondegenerate actions of R2 with toric degree 0 on surfaces.

** Theorem **

1) On S2 there exists a nondegenerate R2 action of toric degree 0, which as 8 orbits of dimension 2. The number 8 is also the minimal number possible (i.e. any action of toric degree 0 will have at least 8 orbits of dimension 2)

2) For any g >= 1, on a closed orientable surface of genus g there exists an action of R2 of toric degree 0 which has exactly 2 orbits of dimension 2. The number 2 is also minimal possible.

Proof.

Existence:

1) Cut S2 into 8 pieces by 3 loops so that each piece is a trigone. According to the normal form theorem ???, there is a action on the closure of a trigone generated by two vector fields X1 & X2 such that X1, X2, and -X1-X2 correspond to the 3 edges. Extend X1 & X2 on the whole S2 by symmetry so that they become invariant wrt the 3 involutions on S2 (each loop is the fixed point set of an involution)

2) In the case of genus >= 1, just need 2 involutions to cut it into four (2g+2)-gones. Construct an action on one of these (2g+2)-gones, and extend it to the whole surface by the reflection principle (so that it becomes invariant with respect to the 2 involutions). Now make the same picture for genus 2g-1, then mod out by the free Z2 action (= product of the 2 involutions) -> get a surface of genus g, and the system on it has only 2 orbits of dimension 2.

Minimality:

1) 1 loop on S2 –> 2 disks with smooth boundary.

2 loops -> there are at least 4 pieces which are 2-gones. Each 2-gone needs to be cut in order to obtain polygones (with at least 3 edges), so in total we have at least 4×2=8 pieces, i.e. 8 orbits of dimension 2 of the action.

2) 2 is minimal, because when one crosses over an edge, the orientation given by X1 wdege X2 changes -> 2 regions which share an edge can not be the same on a orientable surface.

*5.2. Three-dimensional case*

Toric degree 3: T3

Toric degree 2:

1) All singular orbits are non-twisted T2: ambient manifold = orientable T2-fibration over T1, with any monodromy group.

2) 2 singular orbits T1: ambient manifold = any lense space (including the cases S3 and S2 x T1)

3) 2 singular twisted T2: non-orientable. Describe fundamental group ?

4) 1 singular T1 and 1 twisted T2: non-orientable. Orientable 2-leaf covering = ?

Toric degree 1:

Seifert fibrations & raph manifolds ? Rationally elliptic ?

Toric degree 0:

Invariant = minimal number of hyperbolic domains

Question: What is this invariant for S3 ? T3 ? other simple 3-dim manifolds ?

*5.3 Four-dimensional case*

Toric degree = 4: T4

Toric degree = 3: similar classification

Toric degree = 2 -> Results of Orlik and Raymond on possible M4

Toric degree = 1 -> Results of Funtushel on possible M4

Toric degree = 0 –> again minimal number of domains as an interesting invariant. Can say anything ? Any other invariants ?

**6) Final remarks**

???

**References**

Bates & Cushman, What is an integrable non-Hamiltonian system

Camacho, Morse-Smale $R^{2}$-actions on two-manifolds. Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 71–74. Academic Press, New York, 1973.

Chaperon

Lima, Common singualrities of commuting vector fields on 2-manifolds, Comment Math Helv 1964

M Sabatini, Dynamics of commuting systems on two-dimensional manifolds, Annali di Matematica pura ed applicata, 173 (1997), 213-232.

NT Zung, Nondegenerate singularities of integrable non-Hamiltonian systems, arxiv 2011.

(to be added)

**Notes to ourselves:**

* Draw some pictures for illustration:

–> 2-dimensional polygons with hyperbolic R2 actions

–> some isomorphisms (normalizing maps)

* This paper is “elementary” but still quite interesting (and not more elementary than papers by other people on similar subjects). It can get quite long too (>= 20 pages ?)

* Submit to a good journal, e.g. one of the following:

(Avoid Elsevier journals !)

Archive for Rational Mechanics and Analysis

Ergodic Theory and Dynamical Systems

Geometry and Topology

Int Math Res Notices

Journal fuer die Reine und Angewandte Mathematik

Transactions of the American Mathematical Society

What Camacho called Morse-Smale R2-actions on 2-manifolds are the same as our nondegenerate integrable systems of type (2,0), i.e. nondegegerate R2-actions on 2-manifolds.

We rediscover Camacho’s theorem about existence of such systems on any compact 2-manifolds (the case when n=2).

Camacho also obtained some Morse-type inequalities for such systems.

Problem: construct a kind of Morse theory for nondegenerate n-dimensional systems ?

Camacho also showed the existence of degenerate systems which cannot be pertured into nondegenerate one, i.e. the set of all nondegenerate systems is not dens in the space of all systems ?

Hyperbolic domains

O – hyperbolic domain

\bar{O} – its closure

m – a point on O

For each H – orbit of the action \rho in \bar{O}, denote by C_H the set of elements v of the Lie algebra Rn such that the associated flow \phy^t_{X_v} (m) tends to a point on H when t tends to infinity

Proposition:

1) C_H does not depend on the choice of m

2) The falmily (C_H) is a partion of Rn

3) The closure of C_H is a strictly conves polyhedral cone in Rn, and C_H is the “interior” of this cone. The word strict means that the cone doesn’t contain a straight line.

4) dim C_H + dim H = n

5) C_K is a face of C_H if and only if H is a face of K

6) C_O = origin of Rn, and for each (n-1)-1dim face Fi of O the corresponding C_F_i is R_+.v_i, whre v_i is the element of Rn such that X_v_i vanishes on F_i and is locally of the type x d/dx near Fi

(This is the cone decomposition of Rn dual to the cell decomposition of \bar{O} given by the action)

Converse proposition: Any strictly convex cone decomposition of Rn together with a choice of v_i can be realized

The name “elliptic component” may be misleading ?

What would be a better name ?

Maybe elliptic-focus ?

Elbolic?(Elbolic = El-liptic + Hyper-bolic)

Elbolic actions and topological toric manifolds

Elbolic actions = nondegenerate actions of Rn on n-manifolds without hyperbolic singularity.

Proposition: An elbolic action has only one n-dimensional orbit, and the singular set is of codimension (at least) 2.

Let p be a singular point of highest corank in an elbolic action, then the orbit Op through p is a torus T^s, and the n-dimensional orbit is of type T^(m+s) X R^m, with 2m+s = n. In particular, the toric degree of an elbolic action is at least n/2

Consider the special case when there is a fixed point. Then n=2m, and the toric degree is m.

It turns out that elbolic actions with a fixed point correspond to the so called topological toric manifolds recently introcduced by some Japanese mathematicians.

Some references:

Hiroaki Ishida, Yukiko Fukukawa, Mikiya Masuda, Topological toric manifolds, arXiv:1012.1786

(In partiular, they introduced the notion of “topological fans” and showed that There is a one-to-one correspondence between omnioriented topological toric manifolds (omnioriented means that the orientation is fixed on each stratum of the singular set also) and nonsingular complete topological fans.

Li Yu, On Transition Functions of Topological Toric Manifolds, arXiv:1110.4527

Suyoung Choi, Mikiya Masuda, Dong Youp Suh, Rigidity problems in toric topology, a survey, arXiv:1102.1359

Hiroaki Ishida, Invariant stably complex structures on topological toric manifolds, arXiv:1102.4673

A related notion is quasitoric manifolds introduced by Davis-Januszkievicz. A quasitoric manifold is a closed smooth manifold endowed with a locally standard (S1)n-action, whose orbit space is a simple polytope. Theorem of Ishida-Fukukawa-Masuda: The family of topological toric manifolds with the restricted action of the compact torus properly contains that of quasitoric manifolds up to equivariant homeomorphism.

Suyoung Choi (Ajou Univ., Korea)

Title: Invariance of Pontrjagin classes of torus manifolds.

Abstract: A torus manifold of dimension 2n is a closed smooth manifold having a half-dimensional effective torus Tn action with the non-empty fixed point set. A typical example of torus manifolds is CPn with the standard torus action. Petrie has shown that all homotopy equivalence between a homotopy projective space M and CPn preserve their Pontrjagin classes if M is a torus manifold, although Pontrjagin classes are not invariant under homotopy equivalences. In this talk, we discuss about the invariance of Pontrjagin classes of torus manifolds. In general, the Pontrjagin classes of torus manifolds is not preserved by homotopy equivalences. However, we show that, in the case where the cohomology of M is isomorphic to that of a Bott manifold or a product of projective spaces, their Pontjragin classes are preserved by homotopy equivalences, which generalizes the Petries theorem immediately.

Soumen Sarkar (ISI, India)

Title: A class of torus manifolds with nonconvex orbit space

Abstract: We study a class of smooth torus manifolds whose orbit space has the structure of a simple polytope

with holes. We prove that these manifolds have stable almost complex structure and give combinatorial formula

for some of their Hirzebruch genera. They have (invariant) almost complex structure if they admit positive omniorientation.

In dimension four, we calculate their homology groups, construct a symplectic structure on a large

class of these manifolds, and give a family which is symplectic but not complex.

Li Yu (Osaka City Univ. & Nanjing Univ., Japan & China)

Title: On transition functions of topological toric manifolds

Abstract: We show that any topological toric manifold can be covered by finitely many open charts so that all

the transition functions between these charts are Laurent monomials of z j’s and z j’s. In addition, we will describe

toric manifolds and some special class of topological toric manifolds in terms of transition functions of charts up

to (weakly) equivariant diffeomorphism.

http://en.wikipedia.org/wiki/Quasitoric_manifold

Davis, M.; Januskiewicz, T. (1991), “Convex polytopes, Coxeter orbifolds and torus actions”, Duke Mathematical Journal 62 (2): 417–451

Small cover of a simple polytope, mentioned in the paper by Davis & Januskiewicz (1991):

If the polytope has m facets then the group is (Z/2Z)^m. Each elements of the group corresponds to a copie of the cover. The copies glue together in an obvious way into a smooth manifold on which (Z/2Z)^m acts and for which the polytope is a fundamental domain of this action.

Using this small cover construction and the reflection principle, and hyperbolic action on a polytope (with the interior of the polytope as the regular orbit) can be lifted to a hyperbolic action on a manifold, which admits this polytope as a closed hyperbolic domain.

28 pages so far, and not done yet. We are working day and night on it :-)

The goal now is to finish this paper before the end of this week, and to submit it before 04/March.

This paper is taking more time, getting longer, and also more interesting than we first thought :D

Ok the paper is almost finished now, 2-3 weeks behind schedule.

But it is 50+ pages, much longer than originally expected !

What journal should we submit it to ?

Some possible choices :

* Geometry and Topology ?

* J. Diff. Geom. ?

* Acta ? Why not :D