Monthly Archives: June 2014

Contact Hamiltonians (Part I)

This entry follows the post Contact Hamiltonians (Introduction), where we discussed normal forms for contact forms and the appearance of contact Hamiltonians. In this entry we will focus on the 3–dimensional situation and hence we will be able to write formulas and draw (realistic) pictures.

Consider a 2–sphere of radius 1 in the standard tight contact Euclidean space (\mathbb{R}^3,\lambda_{st}=dz+r^2d\theta). Its characteristic foliation (defined by the intersection of the tangent space and the contact distribution) has two elliptic singular points in the north and south poles and all the leaves are open intervals connecting the north and the south pole. Take a transversal segment I=[0,1] connecting the poles (a vertical segment will do). Given a point in the segment we can consider the unique leaf through that point and move around the leaf until we hit the interval I=[0,1] again. This defines a diffeomorphism of the interval [0,1] fixed at the endpoints. We will call this diffeomorphism the monodromy of the foliation (and note that conversely any diffeomorphism will give a foliation on the 2–sphere via a mapping torus construction and collapsing the boundary). This is drawn in the following figure:


In the figure the monodromy map is represented by the orange arrow. This monodromy does not have fixed points (this is crucial). Let us look at the monodromy in the sphere of radius \pi+c , where c is a small positive constant, in the overtwisted contact manifold (\mathbb{R}^3,dz+rtg(r)d\theta). The overtwisted monodromy is drawn in the next figure:


There are 3 types of points in the vertical transverse interval I=[0,1]. The Type 1 points belong to a leaf, Leaf I in the figure, such that the points move down in the segment. The Type 2 points are the points between the unique pair of closed leaves, these belong to Leaf II and move up. The Type 3 points are fixed points, there are two leaves of this type (Leaf III). The monodromy is represented by the blue arrows.

Hence, we can encode the tight and the overtwisted foliations on the 2–sphere in terms of their monodromies in the following figure:


In the last entry we explained a relation between monodromies and contact Hamiltonians. Consider a contact form dz-H(x,y,z)dx in \mathbb{R}^3, this is a quite general normal form (which we can obtain by trivializing along the y–lines of \mathbb{D}^2(x,y)). If we restrict to the sphere x^2+y^2+z^2=R^2 we can write H in terms of H=H(x,z) at points where the implicit function theorem works. Then the characteristic foliation is nothing else than the solution of the time–dependent (x is the time) differential equation dz-Hdx=0 on the interval I=[-1,1] given by the coordinate z. Hence the contact Hamiltonian yields the ODE  to which the monodromy is a solution.

Tool: How do we obtain a piece of a disk in standard contact (\mathbb{R}^3,dz-ydx) with a given characteristic foliation ?

Answer: Consider a disk in the (z,x)–plane and a function H(z,x). The standard contact structure dz-ydx restricts to the graph of H in \mathbb{R}^2(z,x)\times\mathbb{R}(y) as dz-ydx|_{\{y=H\}}=dz-Hdx.

For instance, let us consider the following function H(z) for z=[-1,1]:


This function H can be considered as a function on the polydisk (x,z) which is represented by the lower square in the third figure (the whole figure is PL immersed in the standard contact 3–space). Its image is the bumped square drawn above it, and we may consider the PL sphere obtained by adding the vertical annulus connecting the domain and the graph. The characteristic foliation on the bottom piece is by the horizontal z–lines, on the annulus the foliation is vertical and on the top piece the foliation is drawn on the left. Note that the characteristic foliation in this immersed PL sphere has a closed leaf (in red) coming from the fixed point (or zero, if we look at it horizontally) of H.

Let us briefly focus on the existence of a contact structure in a region bounded by a domain and a graph as in the previous paragraph.

Exercise: Does there exist a contact structure filling the following pink region ?


(The contact structure should restrict to the germs (in purple) already defined on the boundary.)

Answer: Yes. This is already embedded in \mathbb{R}^3, hence we just need to restrict the ambient contact structure. (This should be compared with the previous post where this question was also formulated and answered in terms of the positivity of the function H).

The second exercise we need to solve is as simple as the previous one, let us however draw the figures in order to keep them in mind.

Annulus Problem (weak): Does there exist a contact structure in the (yellow) annulus ?


The contact structure should also restrict to the germs (in purple and green) already defined on the boundary.

Answer: Yes, again this is already embedded in standard contact Euclidean space. This is yet another instance of the relevance of order. If one Hamiltonian is less than another one, then we can obtain a contact structure on the annulus.

This will be formalized in subsequent posts using the notion of domination of Hamiltonians and their corresponding contact shells. We shall not use this language right now.

We are now going to prove Eliashberg’s existence theorem in dimension 3 from the contact Hamiltonian perspective (i.e. from the monodromy viewpoint). The fundamental fact is that we only need to extend contact structures up to contactomorphism and this is translated to the fact the Hamiltonians can be conjugated.

Annulus Problem (strong): Does there exist a contact structure on the following region ?


Answer: If we are able to conjugate the bottom Hamiltonian (in green) strictly below to the upper one (in purple), then we can use the contact structure of the embedded annulus (weak version of the annulus problem). Hence, it all reduces to the order (or rather, the lack thereof).

Fundamental Fact: There exists a conjugation of the bottom Hamiltonian such that it is strictly less than the upper one. In general, given two Hamiltonian with fixed points which are positive at the endpoints of the interval, there exists a conjugation bringing one of them below the other.

(This is an exercise with functions in one variable, in higher dimensions this is no longer simple and this is precisely the main point that M.S. Borman, Y. Eliashberg and E. Murphy have understood).

Let us prove Eliashberg’s 3–dimensional existence theorem, we focus on the extension part (part 2 according to the post three entries ago).

Extension Problem (Version I): Suppose that there exists a contact structure on the complement of a ball B^3 in a 3–fold (which is given by Gromov’s h–principle, see previous posts) and that the characteristic foliation on the boundary S_h^2 has monodromy with fixed points (h stands for hole). Can we extend the contact structure ?

Suppose that there exists a sphere S_{ot}^2 somewhere inside the manifold with an overtwisted monodromy (in blue, see above) in its characteristic foliation. Consider the annulus A_{ot}=S_{ot}^2\times(-\tau,\tau). Use the south poles of S_{ot}^2\times\tau and S_h^2 to connect both and obtain an annulus A such that the monodromy in the exterior boundary sphere is the concatenation of the contactomorphisms of the intervals (green#pink). Hopefully this figure helps:


The monodromies of the foliations in the two spheres bounding the annulus A_{ot} are drawn in pink (exterior boundary) and blue (interior boundary). The monodromy in green is that of S_h^2. Connecting the spheres S_h^2 and S_{ot}^2\times\{\tau\} yields a sphere with the monodromy green#pink (the transition area is purple, this has some relevance but it is not essential). Consider the annulus A bounded by S_h^2\#(S_{ot}^2\times\{\tau\}) and S_{ot}^2\times\{-\tau\}. We have reduced the problem of extending the contact structure to the interior of S_h^2 to the problem of extending the contact structure in the annulus A. In the exterior boundary of A the characteristic foliation is green#pink and on the interior is red (which comes from moving blue).

Extension Problem (Version II): Does there exists a conjugation such that (the graph of) any contactomorphism can be conjugated to lie beneath any other (graph) ?

Answer:  No. Fixed Points are an obstruction. However, if we restrict ourselves to the same question in the class of contactomorphisms with fixed points the answer is yes. This is exactly the Fundamental Fact stated above.

How do we conclude the proof ? Conjugate the red Hamiltonian to lie beneath the green#pink Hamiltonian and use the contact structure in the resulting annulus (as embedded in standard contact space). Assuming Gromov’s h–principle and the technical work in order for the foliation to be controlled, this argument concludes the theorem.

(We have disregarded some details, but the idea of the argument is the one described above. Observe that the parametric version of the existence problem in dimension 3 is quite immediate from the Hamiltonian perspective.)

Note also that we do not need the whole sphere S^2_{ot}: in order to use the argument with the Hamiltonians we can cut the North pole of S^2_{ot} and retain just the remaining disk, which is an overtwisted disk.

There is a substantial advantage in this proof of the 3–dimenisonal case: we can define an overtwisted disk \mathbb{D}^{2n} in higher dimensions 2n+1 to be the object that appears when using the contact Hamiltonian on a simplex \Delta^{2n-1} given by


(We will give precise definitions in the subsequent entries.)

The strategy of the argument works in higher dimensions if we can prove the Fundamental Fact stating that there is enough disorder for contact Hamiltonians. In the next entries we will focus on this crucial step in higher dimensions and conclude existence.


Filed under Uncategorized

Contact Hamiltonians (Introduction)

This entry is part of the series of posts on the recent work of  M. S. Borman, Y. Eliashberg and E. Murphy on the existence and classification of overtwisted contact structures in all dimensions. In the previous two entries the construction in the 3–dimensional case and Gromov’s h–principle for the open case have been explained.

The essential fact in Eliashberg’s 3–dimensional argument (this is part 2, two entries ago) is the control on the characteristic foliation: the extension problem is reduced to being able to fill a 2–sphere with a contact ball given a particular characteristic foliation on the boundary. This can be done explicitly by deforming the local model provided by the standard overtwisted contact ball in order to have the given characteristic foliation on the boundary. The construction in higher dimensions is not quite the same and it relies on the use of contact Hamiltonians, hence this and subsequent entries. This first introductory entry should help the reader to follow the next entries, each entry should however be readable on its own.

Consider a contact manifold of dimension 2n+1. The information of a contact structure is contained in a 1–form (locally this is the data of 2n+1 functions, plus another one as a conformal factor, satisfying 1 non–degeneracy equation). This data can be drastically reduced when restricted to simple topological subsets if we choose appropriate coordinates: for instance, the Darboux theorem tells us that the normal form of a contact 1–form around a point is \alpha=dz-\sum_{n=1}^{2n}y_idx_i. It will be helpful for the reader to understand the geometric proof of the Darboux theorem, see Theorem 2 in Page 5 of Topological methods in 3-dimensional contact geometry. The strategy is finding a suitable flow to obtain the desired coordinates.

Suppose that we have a contact manifold (M,\alpha_0) and a codimension–1 distribution \xi=\ker(\alpha) on the manifold M\times\mathbb{D}^2(r,\theta) such that it restricts to \ker(\alpha_0) on each slice M\times\{pt.\}. In these hypotheses:

Lemma: There exists the following normal form for the 1–form \alpha, we can write \alpha=\alpha_0+H(p,r,\theta)d\theta for some smooth function H:M\times\mathbb{D}^2\longrightarrow\mathbb{R}.

Proof: Consider the product manifold M\times\mathbb{D}^2 as a trivial fibre bundle over the disk \mathbb{D}^2. The data in the hypothesis gives a connection in this bundle whose parallel transport is by contactomorphisms, it is defined as the skew–orthogonal complement of the symplectic subspace \ker(\alpha_0) in the bundle (\xi,d\alpha) with respect to the 2–form d\alpha (which is not necessarily symplectic). We can then consider the radial vector field in the base \mathbb{D}^2 and lift it to the total space M\times\mathbb{D}^2. The pull–back of the contact form by this flow is (conformally) of the form \alpha_0+H(p,r,\theta)d\theta for some function H:M\times\mathbb{D}^2\longrightarrow\mathbb{R}. The reason being that the radial factor dr cannot appear because in the trivializing coordinates (provided by the flow of the lift), the lift of the radial vector field belongs to the distribution. \hfill\Box

There are a couple of technical details regarding the existence of the flow, which can be translated into the size of the base disk. Let us not focus on that. Thanks to the Lemma we have the following reduction of the extension problem.

Suppose that on a given almost contact (2n+1)–fold V we have a contact structure on all of V except on a neighbourhood Op(M)\cong M\times\mathbb{D}^2 of a codimension–2 submanifold M with trivial normal bundle. If the almost contact structure \xi satisfies the hypothesis for the Lemma in Op(M), then the extension problem for the contact structure is reduced to:

Problem: Given a germ of a contact structure on M\times S^1 described by a function H:M\times S^1\times(1-\varepsilon,1+\varepsilon)\longrightarrow\mathbb{R}, does there exist a contact structure on M\times\mathbb{D}^2 such that it restricts to the given germ on M\times S^1 ?

There are two remarks at this point. First, the meaning of the function H is really geometric. It describes the angle of rotation of the contact structure in the radial direction, in particular the condition for \alpha_0+Hd\theta to be a contact structure on M\times S^1\times(1-\varepsilon,1+\varepsilon) reads \partial_r H>0 (this is often stated as the contact structure has to rotate). Second, the extension does not need to be of the form \alpha_0+Hd\theta, we just need a contact structure on M\times\mathbb{D}^2.

Example 1 (Tight): Consider (M,\alpha_0)=((-1,1),dz) and the function H(p,r,\theta)=r^2. The contact form is \alpha=dz+r^2d\theta and since the function H verifies the contact condition on B^3=(-1,1)\times\mathbb{D}^2 this defines a contact structure on B^3. This is the standard contact structure on the ball.

Example 2 (Overtwisted): Consider (M,\alpha_0)=((-1,1),dz) and H(p,r,\theta)=r\cdot tg(r). The contact form is then \alpha=dz+rtg(r)d\theta, which should be read as \alpha=cos(r)dz+rsin(r)d\theta. This is the standard overtwisted contact structure on the ball B^3=(-1,1)\times\mathbb{D}^2 if the radius of the disk is larger than \pi.

This second example has the following very nice feature: the function H(r)=rtg(r) is negative at r=2. This provides a solution to the problem of extending a germ in (-1,1)\times S^1\times\{2\} to the interior (-1,1)\times\mathbb{D}^2 if this germ is everywhere negative. Although a priori it seems non–sense to go from 0 to a negative value growing (in order to preserve the contact condition) this can be done by inserting a pole, i.e. going to infinity (and then continuing from minus infinity). This phenomenon underlies many h–principles, try to solve for instance Section 4.1.1 from Chapter 4 in Eliashberg–Mishachev book.

The functions H appearing in the above constructions are called contact Hamiltonians.

Problem (Easy Case): Given a germ of a contact structure on M\times S^1 described by a positive function H:M\times S^1\times(1-\varepsilon,1+\varepsilon)\longrightarrow\mathbb{R}, does there exist a contact structure on M\times\mathbb{D}^2 such that it restricts to the given germ on M\times S^1 ?

Answer: Yes. In this case the extension can be a contact structure of the form \alpha_0+\widetilde{H}d\theta where \widetilde{H} extends H and is such that \partial_r\widetilde{H}>0. Certainly, we just need to construct a function which at the origin looks like \widetilde{H}=r^2 and then it grows in the radial direction until we reach the value given by H on the boundary M\times S^1. The existence of such a function is immediate. \hfill\Box

The difficult case is that of a germ of a contact structure defined by a Hamiltonian which is negative in some points and positive in others (the presence of such negativity requires overtwistedness). The situation described above is quite hard because we may not even understand the (contact) topology of M. The first step is to focus on M=\Delta^{2n-1} a (2n-1)-ball, or star–shaped domain, in \mathbb{R}^{2n-1}.

In the next entry, Contact Hamiltonians (Part I) we will continue to use contact Hamiltonians and relate them to Eliashberg’s 3–dimensional argument using the characteristic foliation. The essential word will be monodromy.

In the context above, monodromy arises as follows: consider the contact germ on M\times S^1(\theta) and lift the vector field \partial_\theta to the connection defined before. Its flow at time equal to the length of the circle (say 1) defines a contactomorphism of the fibre M\times\{0\}. This is the monodromy contactomorphism.

There is however another way to obtain a contactomorphism of (M,\alpha_0) if we have a  function H:M\times S^1\longrightarrow\mathbb{R} (referred to as a time–dependent contact Hamiltonian). Indeed, compute the Hamiltonian contact vector field X associated to H, which is the unique solution of

\alpha_0(X_\theta)=H_\theta and d\alpha_0(X_\theta,\cdot)=-dH_\theta+dH(R_{\alpha_0})\cdot\alpha_0

where R_{\alpha_0} is the Reeb vector field. Then the time–1 flow of the Hamiltonian vector field is a contactomorphism of M. This contactomorphism is said to be generated by the contact Hamiltonian H.

Lemma: Given the contact germ \alpha_0+Hd\theta on M\times S^1, the monodromy contactomorphism coincides with the contactomorphism generated by H.

The proof of this lemma is a nice exercise on linear algebra using the defining equation of the connection. This setup can be explicitly studied in 3–dimensions where the monodromies (and the functions H) can be drawn and they correspond to ODEs in the plane. In the next post we will proof Eliashberg’s theorem in dimension 3 from the contact Hamiltonian perspective.

Leave a comment

Filed under Uncategorized

Gromov’s h-principle for open contact manifolds

Continuing towards a discussion of the proof of existence and classification of overtwisted contact structures in higher dimensions, here I want to talk about h-principles, and contact structures on open manifolds.


Given a partial differential equation or partial differential relation (like the contact condition \alpha \wedge d\alpha > 0), one can formally replace the derivatives of the variables with independent formal variables (i.e. \alpha \wedge \eta >0 for a 2-form \eta). Solving this new problem where the derivatives are replaced by independent formal variables is purely an algebraic topology problem. If the algebraic topology problem has no solution then certainly the partial differential relation has no solution. However, it is generally surprising when the converse holds: namely, the existence of a solution to the algebraic problem implies the existence of a solution of the partial differential relation. A theorem that proves this type of statement is referred to as an h-principle.

The language of jet bundles and holonomic sections will help make this more precise below.

The main results of Borman, Eliashberg, and Murphy that we are heading towards are an h-principle for contact and almost contact structures on higher dimensional closed manifolds which says that any almost contact structure is homotopic through almost contact structures to an actual (overtwisted) contact structure, and a parametric version of this h-principle which says that any family of almost contact structures connecting two genuine overtwisted contact structures can be homotoped to a family of genuine contact structures connecting the fixed overtwisted contact structures on the ends.

While contact structures on closed manifolds can have incredibly complicated classifications (because of the rigidity of tight contact structures), it is a result of Gromov that on open manifolds the geometric subtlety disappears and the classification of contact structures is reduced to algebraic topology by an h-principle. This post is based on a talk given by Kyler as part of the discussion of the proof of flexibility of overtwisted contact structures in higher dimensions, though the original source for the content is Gromov’s Partial Differential Relations book.

Define a (cooriented) almost contact structure on an odd dimensional manifold to be a cooriented hyperplane distribution, together with a non-degenerate 2-form on the distribution. In dimension 3, this is homotopy equivalent to the space of co-oriented 2-plane distributions. Gromov’s theorem is:

Let V be an open manifold. Then the inclusion of cooriented contact structures on V into cooriented almost contact structures on V is a homotopy equivalence.

The proof is based on two main ideas: the holonomic approximation theorem on neighborhoods of codimension one polyhedra, and the fact that all open smooth manifolds smoothly retract onto a neighborhood of a complex of codimension at least one. I’ll start with the former.

The 1-jet space of a fiber bundle X\to V, is a bundle J^1(X)\to V where the fiber over p \in V consists of sections of X defined over a neighborhood of p up to an equivalence which equates sections that agree up to 1st order near p. (The r-jet bundle is defined similarly where you equate sections which agree up to rth order, but here we will only need the 1-jet bundle.) A section of J^1(X)\to V chooses an equivalence class of sections over each point in V: for each p\in V, s(p)=(f(p),\alpha(p)) where f(p) is a point in the fiber X_p, and \alpha(p) specifies the first partial derivatives of a function at that point. However, even though the section is smooth, \alpha(p) need not specify the actual derivative of f(p) since \alpha(p) is encoded as an independent direction in the fibers of J^1(X). A holonomic section of a 1-jet space is one where this linear variation specified by \alpha(p) agrees with the actual partial derivatives of the differentiable section of X\to V given by the 0th order information of the section. The holonomic approximation theorem aims to approximate an arbitrary section of the 1-jet bundle by a holonomic section as well as possible.


Here the blue curve represents a section of J^1(X). The grey curve represents its projection to the 0th order information, and the 1st order information is encoded in the dimension coming out of the page. Representing the value the blue curve takes in this dimension by a green line of the appropriate slope centered at each point on the grey curve, we see that this is not a holonomic section because the 1st order information is not tangent to the curve.

The important relevant example for Gromov’s theorem is when X=\Lambda^1(V), so sections of the bundle are 1-forms. Sections of the 1-jet space keep track of two coordinates: the pointwise values of the underlying 1-form and its formal linear variation. Locally, \Lambda^1(V) is a trivial bundle, and a section is just the graph of a function on U\subset V. Modding out by the equivalence relation, we get that for a section s:V\to J^1(\Lambda^1(V)), s(p) keeps track of the point p, a point in T^*_p(V)=\Lambda^1_p(V) and an n by n matrix at that point which specifies the formal first partial derivatives of a graph in that equivalence class (where n is the dimension of V). Symmetrizing this matrix (A-A^T), gives the coefficients for a 2-form. When the section of J^1(\Lambda^1(V)) is a holonomic section, this 2-form built from the 1st order information of the section, is the exterior derivative of the 1-form which gives the 0th order information of the section. Given any pair (\alpha, \beta) of a 1-form and a 2-form, there is a section of J^1(\Lambda^1(V)) such that the 0th order information gives \alpha and after symmetrizing the 1st order information we get \beta. For holonomic sections, this process gives a pair (\alpha, d\alpha) where \alpha is a 1-form.

There are certain limitations on the extent to which we can approximate an arbitrary section by a holonomic one. For example if we consider the 1-jet space of the bundle \pi_1: \mathbb{R}\times \mathbb{R}\to \mathbb{R}. A section of the 1-jet space is given by specifying the pointwise data and a formal 1st derivative. An example of a section has pointwise data given by the graph of f(x)=x, and formal derivative specified as 0 (horizontal lines at each point). To approximate this by a holonomic section, we would need to find a function g whose pointwise values only differ from those of f(x)=x by \varepsilon, and whose derivative only differs from zero by \varepsilon. Such a function would contradict the mean value theorem. So we cannot hope to approximate an arbitrary section by a holonomic one at every point. On the other hand, we can approximate the section in a small neighborhood of a point.

This motivates the idea to look at codimension 1 subspaces. Taking the previous example and just taking the product with a trivial extra dimension with coordinate y, we run into the same problem: that if we can only move up with a tiny slope in the x-direction, we cannot get up far enough by just moving along a path that has slope 1 in the x direction and does not move in the y-direction. However if we are allowed to perturb the path to lengthen it in the extra y-dimension that we have by adding many zig-zags, then we can do this approximation.


Moving along the black curve, there is no holonomic approximation which stays close to the horizontal planes. However, moving along the perturbed red curve, we can find a closer approximation which is holonomic.

This leads us to the precise theorem:

Holonomic approximation theorem: Let A \subset V be a polyhedrong of codimension at least 1 and suppose we have a section of the jet bundle defined over a neighborhood of A. Then for any \varepsilon>0, there is a \varepsilon small isotopy h_t of A (measured in the C^0 topology), and a holonomic section defined in a smaller neighborhood of h_1(A) which is \varepsilon close to the chosen section.

Suppose we have an almost contact structure (\alpha, \eta) on the open manifold V of dimension n. In order to use this theorem to prove Gromov’s theorem, we must identify a good codimension 1 subset of our open manifold V, where we can use holonomic approximation to find a genuine contact structure on a neighborhood of this subset which is \varepsilon close to the almost contact structure we are considering. Choose a triangulation of V, and for each top dimensional simplex, choose a path from the barycenter of that simplex out to infinity which avoids the barycenters of other simplices. The parts of the 2-skeleton which do not intersect these paths form a codimension 1 subcomplex S. The entire manifold smoothly deformation retracts onto arbitrarily small neighborhoods of S.

Now apply the holonomic approximation theorem to the pair (\alpha, \eta) (which corresponds to a section of J^1(\Lambda^1(V))) along S. Then on a tiny perturbation of S, there is an actual holonomic section corresponding to (\widetilde{\alpha},d\widetilde{\alpha}) which is very close to (\alpha,\beta). By choosing our \varepsilon sufficiently small so that (\alpha, \beta) and (\widetilde{\alpha},d\widetilde{\alpha}) are sufficiently close, we can ensure that the straight line homotopy between them stays in the space of almost contact structures (since the almost contact condition is an open condition (\alpha\wedge \eta>0). Therefore the holonomic approximation theorem implies we can homotope our almost contact structure to be contact on a neighborhood of the perturbed S.

Observe that if g_1:V\to V is the end of the deformation retraction which sends V into the neighborhood of S where the almost contact structure is now genuinely contact, then g_1 pulls back the almost contact structure on V to a genuine contact structure on V. The deformation retraction provides a homotopy between the almost contact structure which is contact on the neighborhood of S to the genuine contact structure coming from this pullback. Therefore concatenating the homotopy provided by the holonomic approximation theorem with the homotopy provided by the deformation retract, gives a homotopy from our original almost contact structure to an actual contact structure on V.

Leave a comment

Filed under Uncategorized