On normal Darboux coordinates

Boris Fedosov

Dedicated to Professor M.I. Vishik on the occasion of his eightieth birthday

Abstract. For a given symplectic connection we try to find Darboux local coordinates for which connection coefficients take the simplest form, the socalled normal Darboux coordinates. To this end we introduce a nonlinear Cayley transform which gives a convenient parametrization of local symplectomorphisms and prove the existence and uniqueness of such coordinates on the level of jets. At the same time we construct a counterexample showing non-correctness of the problem of finding normal Darboux coordinates.

1. Introduction

Recall the notion of normal local coordinates on a manifold with an affine connection. Let M be a smooth manifold with an affine torsion-free connection ∂ and x^{1}, x^{2}, …, x^{n} be local coordinates with the origin x_{0} ∈ M. We call them normal or geodesic coordinates with respect to the connection ∂ if the following identity holds

Γ_{jk}^{i}(x)x^{j}x^{k} ≡ 0 (1.1)

where Γ_{jk}^{i} are connection coefficients (Christoffel symbols) of ∂ in a given coordinate chart. This identity has an invariant meaning: the rays emanating from the origin are geodesics, that is

∂_{ẋ(t)}ẋ(t) = ẍ^{i} + Γ_{jk}^{i}ẋ^{j}ẋ^{k} = 0. (1.2)

Here x^{i}(t) = x^{i}t, t ∈ [0, ∞) is a ray through the point x = {x^{1}, x^{2}, …, x^{n}} ≠ x_{0}. Thus, for the rays we have ẍ^{i} = 0, ẋ^{i} = x^{i}, so that (1.2) reduces to (1.1).

The theory of ordinary differential equations applied to (1.2) gives the following existence theorem.

Theorem 1.1. For any x_{0} ∈ M there exists a neighborhood O and a normal local coordinate system on O with the origin x_{0}. This system is unique up to linear change of variables.

Thus, at any point x_{0} ∈ M we have essentially unique local coordinate system for which connection coefficients take the simplest form (1.1). Normal coordinates play an important role in local differential-geometrical computations.

2000 Mathematics Subject Classification. 53D05, 53D22.

The object of our interest in this paper is a symplectic connection ∂ on a symplectic manifold (M, ω). A detailed treatment of symplectic connections may be found in [GRS]. We consider Darboux local coordinates x^{1}, x^{2}, …, x^{2n} on M for which the coefficients of the symplectic form ω = 1/2ω_{ij}dx^{i} ∧ dx^{j} are constant. It is known (see, e.g. [GRS]) that the connection coefficients

Γ_{ijk} = ω_{im}Γ_{jk}^{m}

are completely symmetric in i, j, k. Indeed, the symmetry in j, k follows from the vanishing of the torsion and the symmetry in i, k reflects the fact that ∂ preserves the tensor ω_{ij}. We are looking for the local Darboux coordinates for which the coefficients Γ_{ijk}(x) have the simplest form similar to (1.1).

Definition 1.2. Local Darboux coordinates are called normal if the following identity holds

Γ_{ijk}(x)x^{i}x^{j}x^{k} ≡ 0. (1.3)

To avoid confusion, we use the name geodesic coordinates for those satisfying (1.1) reserving the name normal Darboux coordinates for those satisfying (1.3). Note, that geodesic coordinates for a given symplectic connection are not Darboux ones in general, so they are not normal coordinates in the sense of Definition 1.2, although they satisfy (1.3).

By analogy with Theorem 1.1 we formulate the following conjecture.

Conjecture 1.3. For a given symplectic connection ∂ and any x_{0} ∈ M there exists a local normal Darboux coordinate system with the origin at x_{0}. It is unique up to linear symplectic change of variables.

In [F, Chapter 2] a weaker version was proved.

Theorem 1.4. For any x_{0} ∈ M there exists a local Darboux coordinate system with the origin at x_{0} such that the function (1.3) has a zero of infinite order at the origin, that is

Γ_{ijk}(x)x^{i}x^{j}x^{k} = O(|x|^{∞}). (1.4)

Any two such systems differ by a linear symplectic change of variables up to infinite order. More precisely, if x_{1} = {x_{1}^{1}, x_{1}^{2}, …, x_{1}^{2n}} and x_{2} = {x_{2}^{1}, x_{2}^{2}, …, x_{2}^{2n}} are such two systems, then there exists a matrix U ∈ Sp(2n) such that

x_{2} − Ux_{1} = O(|x_{1}|^{∞}).

We will use the term week normal Darboux coordinates if they satisfy (1.4). Theorem 1.4 means that the conjecture is true on the level of formal Taylor series. For real analytic data this theorem gives a real analytic symplectic transformation to normal Darboux coordinates, provided the formal series are convergent. So, our conjecture resembles the Newlander-Nirenberg theorem on integrable almost complex structures [NN].

The notion of normal Darboux coordinates was introduced in the book [F] for the purposes of deformation quantization. For these applications Theorem 1.1 was sufficient. But the conjecture remains a challenging problem which I tried to solve many times in the affirmative but in vain. Each time it became more evident that the conjecture should be false.

The main purpose of this paper is to present a counterexample. More precisely, we show that the linearized problem has no smooth solution in any neighborhood of a given point x_{0}. Strictly speaking, this is not a direct counterexample for the Conjecture 1.3. Our example means that the solution (if exists) has no differential with respect to parameters. In other words, the problem of finding normal Darboux coordinates is non-correct.

Our considerations are purely local, so we deal with the symplectic space ℝ^{2n} with a standard symplectic form ω = 1/2ω_{ij}dx^{i} ∧ dx^{j}, ω_{ij} = const. The main technical tool is a special parametrization of local symplectomorphisms which we call non-linear Cayley transform. Its description is given is Section 2. In Section 3 we use it to prove Theorem 1.4 and in Section 4 give a standard application of this theorem. All this is a necessary background to the last Section 5 where the counterexample is constructed.

I would like to thank A.Weinstein for a stimulating discussion. In particular, he suggested to apply the so-called Moser trick to prove the conjecture. Having thought this application over, I came to a guess that the conjecture might be false.

2. Non-linear Cayley transform

The Cayley transform

U = (1 − A)(1 + A)^{−1}; A = (1 − U)(1 + U)^{−1} (2.1)

gives a convenient rational parametrization of symplectic matrices U ∈ Sp(2n) by means of infinitesimally symplectic matrices A ∈ sp(2n). It is defined, in particular, for A sufficiently close to the identity matrix 1 and gives a one-to-one correspondence between a neighborhood of 0 in sp(2n) and a neighborhood of 1 in Sp(2n). More detail on Cayley transform can be found in the book [W]. Since infinitesimally symplectic matrix A = (a_{j}^{k}) corresponds to a symmetric matrix

S = (s_{ij}) = (ω_{ik}a_{j}^{k}),

the Cayley transform may be viewed as a correspondence between quadratic forms S = 1/2s_{ij}x^{i}x^{j} (we denote the quadratic form and the corresponding symmetric matrix by the same letters) and linear symplectic transformations U ∈ Sp(2n).

Consider now non-linear symplectomorphisms y = f(x) of a neighborhood of 0 ∈ ℝ^{2n} which are sufficiently close to the identity map. We always assume that f(0) = 0. Introduce new variables

z = \frac{x + y}{2} = \frac{x + f(x)}{2} (2.2)

and consider a one-form

ω (\frac{y − x}{2}, dz) = \frac{1}{4}ω_{ij}(y − x)^{i}d(x + y)^{j} (2.3)

Lemma 2.1. The form (2.3) is closed.

Proof. The differential is proportional to

ω(dy − dx, dx + dy) = ω(dy, dy) − ω(dx, dx) + ω(dy, dx) − ω(dx, dy).

The first two summands on the right give 0 since ω is preserved by symplectomorphisms. The last two summands also give 0 because ω(dy, dx) = ω(dx, dy). Indeed, transposing dy and dx, we have to change the sign twice: because ω is skew-symmetric and because exterior differentials anticommute.

We introduce a generating function S(z) of the symplectomorphism f by the identity

dS(z) = ω (\frac{y − x}{2}, dz) (2.4)

or

\frac{x − y}{2} = ∇S(z) (2.5)

where ∇S means the symplectic gradient of S, that is, the vector with coordinates

∇^{i}S = ω^{ij} \frac{∂S}{∂z^{j}} .

From (2.2) and (2.5) we obtain

x = z + ∇S(z), y = z − ∇S(z) (2.6)

which gives a parametrization of symplectomorphisms f by generating functions S(z). The correspondence is one-to-one if we set S(0) = 0. Of course, it is defined if x can be expressed from (2.2) as an implicit function of z (respectively, z can be expressed from the first equation (2.6) as an implicit function of x). In this sense f must be sufficiently close to the identity map or S(z) must be sufficiently small. Note that the condition f(0) = 0 implies that ∇S = 0, so S has a second order zero at the origin. It is easy to see that for quadratic forms S = 1/2s_{ij}z^{i}z^{j} we obtain the linear Cayley transform (2.1). Obviously, we also have a linear Cayley transform between the differential df of a symplectomorphism and the second differential d^{2}S of the generating function. For these reasons we call the correspondence given by (2.6) the non-linear Cayley transform and the function S(z) the Cayley generating function.

Generating functions of symplectomorphisms have a wide field of applications, e.g., in the theory of Fourier integral operators (FIO). Yet our version differs from those used in FIO. Indeed, the latter require a separation of variables into coordinates and momenta, while in our version all the variables are treated on equal footing. This drastically simplifies the formulas making them more manageable.

We finish this section by indicating the relations between Hamiltonian flows and Cayley generating functions. Let f_{t}(x) be a Hamiltonian flow on ℝ^{2n} generated by a Hamiltonian vector field

X^{i}(x, t) = ω^{ij} \frac{∂H(x, t)}{∂x^{j}} = ∇^{i}H.

Lemma 2.2. The generating function S_{T} (z) of the symplectomorphism f_{T} (x) at a point

z = \frac{x + f_{T} (x)}{2}

is equal to the action along the trajectory f_{t}(x), t ∈ [0, T], more precisely,

S_{T} (z) = \frac{1}{2} ∫_{0}^{T} \{\frac{1}{2}ω(f_{t}(x) − x, ḟ_{t}(x)) − H(f_{t}(x), t)\} dt. (2.7)

ON NORMAL DARBOUX COORDINATES 5

Proof. Fixing T and differentiating (2.7), we obtain

dS_{T} = \frac{1}{2} ∫_{0}^{T} \{\frac{1}{2}ω(f_{t}(x) − x, dḟ_{t},) + \frac{1}{2}ω(df_{t} − dx, ḟ_{t}(x)) − \frac{∂H}{∂x^{i}} df_{t}^{i}\} dt

= \frac{1}{4}ω (f_{t} − x, df_{t})|_{0}^{T} + \frac{1}{2} ∫_{0}^{T} \{−\frac{1}{2}ω(dx, ḟ_{t},) + ω(df_{t}, ḟ_{t}) − \frac{∂H}{∂x^{i}} df_{t}^{i}\} dt

= \frac{1}{4}ω(f_{T} − x, df_{T}) + \frac{1}{4}ω(f_{T} − x, dx).

The last two terms under the integral sign vanish since

ḟ^{i} = ω^{ij} \frac{∂H}{∂x^{j}} ,

so that

dS_{T} (z) = \frac{1}{4}ω(f_{T} (x) − x, d(f_{T} (x) + x))

which coincides with (2.4).

Finally, the Hamilton-Jacobi equation for the action S_{T} (z) takes the following form.

Lemma 2.3. As the function of z and T, S_{T} (z) satisfies the equation

\frac{∂S_{T}}{∂T} + \frac{1}{2}H(z − ∇S_{T} (z), T) = 0. (2.8)

Proof. Taking x, T as independent variables, we have

z = \frac{x + f_{T} (x)}{2}.

Thus,

\frac{dS_{T}}{dT} = \frac{∂S_{T}}{∂T} + \frac{∂S_{T}}{∂z^{i}} \frac{ḟ_{T}}{2} ,

and, substituting ∂S_{T} /∂z^{i} from (2.4), we can write further

\frac{dS_{T}}{dT} = \frac{∂S_{T}}{∂T} + \frac{1}{4}ω(f_{T} − x, ḟ_{T}).

On the other hand, it follows from (2.7)

\frac{dS_{T}}{dT} = \frac{1}{4}ω(f_{T} − x, ḟ_{T}) − \frac{1}{2}H(f_{T} , T).

Equating these two expressions for dS_{T} /dT, we obtain

\frac{∂S_{T}}{∂T} + \frac{1}{2}H(f_{T} , T) = 0,

and Equation (2.8) follows since

f_{T} = z − ∇S_{T} (z).

3. Cayley generating function for normal Darboux coordinates

Consider two Darboux charts x_{1} = {x_{1}^{1}, x_{1}^{2}, …, x_{1}^{2n}} and x_{2} = {x_{2}^{1}, x_{2}^{2}, …, x_{2}^{2n}} with the same origin. They are connected by a symplectomorphism x_{2} = f(x_{1}) such that f(0) = 0. We will look for symplectomorphisms from a given Darboux coordinate system x_{1} to a normal Darboux coordinate system x_{2}. First of all observe that the Equation (1.3) is invariant under linear symplectomorphisms. This means that we can normalize f by the condition df(0) = 1. Under this normalization f may be defined in a neighborhood of the origin by means of a generating function S(z) such that

S(0) = 0, dS(0) = 0, d^{2}S(0) = 0,

that is, S(z) has the third-order zero at the origin. We are going to derive a differential equation for S(z).

Let Γ_{ijk}^{1}(x_{1}), Γ_{ijk}^{2}(x_{2}) be connection coefficients of a symplectic connection ∂ in Darboux coordinates x_{1} and x_{2}. Set

z = \frac{x_{1} + x_{2}}{2} = \frac{x_{1} + f(x_{1})}{2}.

Since df(0) = 1, this equation defines x_{1} as an implicit function x_{1} = x_{1}(z) and x_{2} = x_{2}(z) = 2z − x_{1}(z).

Proposition 3.1. The following transition formula holds

Γ_{ijk}^{2}(x_{2})\frac{∂x_{2}^{i}}{∂z^{p}} \frac{∂x_{2}^{j}}{∂z^{q}} \frac{∂x_{2}^{k}}{∂z^{r}} = Γ_{ijk}^{1}(x_{1})\frac{∂x_{1}^{i}}{∂z^{p}} \frac{∂x_{1}^{j}}{∂z^{q}} \frac{∂x_{1}^{k}}{∂z^{r}} + 2\frac{∂^{3}S}{∂z^{p}∂z^{q}∂z^{r}} . (3.1)

Proof. We start with the usual transition formula for connection coefficients from coordinates x_{1} or x_{2} (denoted simply x for short) to coordinates z. Let Γ_{jk}^{i}(x) and Γ̃_{jk}^{i}(z) be corresponding connection coefficients. Then

\frac{∂x^{l}}{∂z^{m}} Γ̃_{jk}^{m}(z) = Γ_{qr}^{l}(x)\frac{∂x^{q}}{∂z^{j}} \frac{∂x^{r}}{∂z^{k}} + \frac{∂^{2}x^{l}}{∂z^{j}∂z^{k}} .

Multiplying by ∂x^{p}/∂z^{i}ω_{pl} and using the relation

\frac{∂x^{p}}{∂z^{i}} ω_{pl} \frac{∂x^{l}}{∂z^{m}} = ω_{im}(z)

for coefficients of the symplectic form, we obtain

Γ̃_{ijk}(z) = Γ_{pqr}(x)\frac{∂x^{p}}{∂z^{i}} \frac{∂x^{q}}{∂z^{j}} \frac{∂x^{r}}{∂z^{k}} + ω_{pl} \frac{∂x^{p}}{∂z^{i}} \frac{∂^{2}x^{l}}{∂z^{j}∂z^{k}} .

Let us write this formula taking x = x_{1} or x = x_{2}. Then by (2.6) x_{1,2} = z ± ∇S, so

\frac{∂^{2}x_{1,2}^{p}}{∂z^{j}∂z^{k}} = ±ω^{lm} \frac{∂^{3}S}{∂z^{m}∂z^{j}∂z^{k}} .

Equating the right-hand sides, we get

Γ_{pqr}^{2}(x_{2})\frac{∂x_{2}^{p}}{∂z^{i}} \frac{∂x_{2}^{q}}{∂z^{j}} \frac{∂x_{2}^{r}}{∂z^{k}} − \frac{∂x_{2}^{m}}{∂z^{i}} \frac{∂^{3}S}{∂z^{m}∂z^{j}∂z^{k}}

= Γ_{pqr}^{1}(x_{1})\frac{∂x_{1}^{p}}{∂z^{i}} \frac{∂x_{1}^{q}}{∂z^{j}} \frac{∂x_{1}^{r}}{∂z^{k}} + \frac{∂x_{1}^{m}}{∂z^{i}} \frac{∂^{3}S}{∂z^{m}∂z^{j}∂z^{k}} .

Taking into account that x_{1}^{m} + x_{2}^{m} = 2z^{m}, we come to (3.1).

Thus, we can express Γ^{2} as

Γ_{ijk}^{2}(x_{2}) (3.2)

= Γ_{pqr}^{1}(x_{1})\frac{∂x_{1}^{p}}{∂x_{2}^{i}} \frac{∂x_{1}^{q}}{∂x_{2}^{j}} \frac{∂x_{1}^{r}}{∂x_{2}^{k}} + 2\frac{∂^{3}S}{∂z^{p}∂z^{q}∂z^{r}} \frac{∂z^{p}}{∂x_{2}^{i}} \frac{∂z^{q}}{∂x_{2}^{j}} \frac{∂z^{r}}{∂x_{2}^{k}}

and further

Γ_{ijk}^{2}(x_{2})x_{2}^{i}x_{2}^{j}x_{2}^{k} (3.3)

= Γ_{pqr}^{1}(x_{1})\frac{∂x_{1}^{p}}{∂x_{2}^{i}} \frac{∂x_{1}^{q}}{∂x_{2}^{j}} \frac{∂x_{1}^{r}}{∂x_{2}^{k}} x_{2}^{i}x_{2}^{j}x_{2}^{k} + 2\frac{∂^{3}S}{∂z^{p}∂z^{q}∂z^{r}} \frac{∂z^{p}}{∂x_{2}^{i}} \frac{∂z^{q}}{∂x_{2}^{j}} \frac{∂z^{r}}{∂x_{2}^{k}} x_{2}^{i}x_{2}^{j}x_{2}^{k},

where

x_{1} = z + ∇S(z); x_{2} = z − ∇S(z). (3.4)

Equating (3.3) to 0, we come to a differential equation for the generating function of the symplectic transformation to normal Darboux coordinates

2\frac{∂^{3}S}{∂z^{i}∂z^{j}∂z^{k}} b^{i}b^{j}b^{k} + Γ_{ijk}^{1}(z + ∇S)a^{i}a^{j}a^{k} = 0 (3.5)

where a and b are vector fields given by the formulas

a = \frac{∂(z + ∇S)}{∂z} (\frac{∂(z − ∇S)}{∂z})^{−1} (z − ∇S) (3.6)

b = (\frac{∂(z − ∇S)}{∂z})^{−1} (z − ∇S). (3.7)

Thus, we come to a non-linear third order differential equation having degeneracy at the origin. The solution S gives a Cayley generating function for a symplectomorphism f to normal Darboux coordinates x_{2}. Unfortunately, we can not solve this equation, so we replace it by a more simple one

2\frac{∂^{3}S}{∂z^{i}∂z^{j}∂z^{k}} z^{i}z^{j}z^{k} + Γ_{ijk}^{1}(z)z^{i}z^{j}z^{k} = 0 (3.8)

obtained from (3.5), (3.6) and (3.7) by dropping ∇S everywhere.

Lemma 3.2. Equation (3.8) has a unique solution with the third order zero at the origin

S(z) = −\frac{1}{4} ∫_{0}^{1}(1 − t)^{2}Γ_{ijk}^{1}(tz)z^{i}z^{j}z^{k}dt. (3.9)

Proof. Consider equality (3.8) on a ray zt, t ≥ 0. It may be written as

2\frac{d^{3}S(tz)}{dt^{3}} + Γ_{ijk}^{1}(tz)z^{i}z^{j}z^{k} = 0.

Integrating three times with respect to t and using that S(tz) has a third order zero at t = 0, we come to (3.9).

We are able now to prove Theorem 1.4.

Proof. Let us start with a given Darboux coordinate system x_{1} = {x_{1}^{1}, x_{1}^{2}, …, x_{1}^{2n}} and corresponding Christoffel symbols Γ_{ijk}^{1}(x_{1}). Using (3.9), define the Cayley generating function S_{1}(z) and the corresponding symplectomorphism f_{1} by the non-linear Cayley transform (3.4). We obtain new Darboux coordinates x_{2} = f_{1}(x_{1}) = {x_{2}^{1}, x_{2}^{2}, …, x_{2}^{2n}} and new Christoffel symbols Γ_{ijk}^{2}(x) given by (3.2). Next, repeat this procedure taking x_{2} and Γ_{ijk}^{2}(x_{2}) as starting data and find S_{2}(z), f_{2} and so on. At the N-th step we get

x_{N}, Γ_{ijk}^{N}(x_{N}), S_{N}(z), f_{N},

where x_{N} = {x_{N}^{1}, x_{N}^{2}, …, x_{N}^{2n}} is a Darboux coordinate system, Γ^{N} are Christoffel symbols of the symplectic connection ∂ for these coordinates, S_{N} the Cayley generating function and f_{N} is a symplectomorphism to the next Darboux coordinates x_{N+1} = f_{N}(x_{N}). Finally, the composition

F_{N} = f_{N} ∘ f_{N−1}, ∘ ⋯ ∘ f_{1}

gives a symplectomorphism from the original coordinates x_{1} to x_{N+1}.

Using induction, let us show that

Γ_{ijk}^{N}(x_{N})x_{N}^{i}x_{N}^{j}x_{N}^{k} = O(|x_{1}|^{N+2}). (3.10)

It is evident for N = 1. Supposing that (3.10) is true and using (3.9) for S_{N}(z) with Γ^{1} replaced by Γ^{N}, we find

S_{N}(z) = O(|z|^{N+2}), ∇S_{N}(z) = O(|z|^{N+1}).

To estimate Γ_{ijk}^{N+1}(x_{N+1})x_{N+1}^{i}x_{N+1}^{j}x_{N+1}^{k}, we use (3.3) with x_{1} replaced by x_{N}, x_{2} by x_{N+1}, S by S_{N} and Γ_{ijk}^{1} by Γ_{ijk}^{N}. Thus,

Γ_{ijk}^{N+1}(x_{N+1})x_{N+1}^{i}x_{N+1}^{j}x_{N+1}^{k}

= Γ_{pqr}^{N}(z + ∇S_{N})a_{N}^{p}a_{N}^{q}a_{N}^{r} + 2\frac{∂^{3}S_{N}(z)}{∂z^{p}∂z^{q}∂z^{r}} b_{N}^{p}b_{N}^{q}b_{N}^{r} (3.11)

where, similarly to (3.6), (3.7),

a_{N} = \frac{∂(z + ∇S_{N})}{∂z} (\frac{∂(z − ∇S_{N})}{∂z})^{−1} (z − ∇S_{N}) (3.12)

b_{N} = \frac{∂(z − ∇S_{N})^{−1}}{∂z} (z − ∇S_{N}) (3.13)

and x_{N+1} = z − ∇S_{N}(z) = z + O(|z|^{N+1}). Introduce a function

φ(τ) = Γ_{pqr}^{N}(z + τ∇S)a_{N}^{p}(τ)a_{N}^{q}(τ)a_{N}^{r}(τ) + 2\frac{∂^{3}S_{N}(z)}{∂z^{p}∂z^{q}∂z^{r}} b_{N}^{p}(τ)b_{N}^{q}(τ)b_{N}^{r}(τ)

where a_{N}(τ), b_{N}(τ) are defined by (3.12), (3.13) with ∇S_{N} replaced by τ∇S_{N}. Then the right-hand side of (3.11) is

φ(1) = φ(0) + (φ(1) − φ(0)) = (φ(1) − φ(0))

since

φ(0) = Γ_{pqr}^{N}(z)z^{p}z^{q}z^{r} + 2\frac{∂^{3}S_{N}}{∂z^{p}z^{q}z^{r}} z^{p}z^{q}z^{r} = 0

by construction. Thus,

φ(1) = φ(1) − φ(0) = O(∇S_{N}) = O(|z|^{N+1}),

proving the desired estimate for the left-hand side of (3.11).

The sequence x_{N+1} = f_{N}(x_{N}) = F_{N}(x_{1}) of symplectomorphisms has the property that for any fixed k the sequence of differentials

d^{k}x_{N+1}|_{x_{1}=0} = d^{k}F_{N}(0) (3.14)

stabilizes for N large enough. Indeed, we see by induction that x_{N} = O(|x_{1}|) and further

d^{k}x_{N+1} = d^{k}(x_{N} + O(|x_{N}|^{N+1})) = d^{k}x_{N} + d^{k}O(|x_{1}|^{N+1})

where the second term vanishes at the origin for N + 1 larger than k. To complete the proof, it remains to show that there exists a symplectomorphism F (x_{1}) whose k-th order differential d^{k}F (0) at the origin coincides with the stabilized value of (3.14). Passing from symplectomorphisms F_{N}(x_{1}) to their generating functions, we see that this problem is equivalent to the well-known construction of a function with the given Taylor expansion at the origin.

We would like to emphasize that this proof does not mean the convergence of the sequence F_{N}(x_{1}) in a fixed neighborhood of the origin in some norm.

4. Application

Here we describe a standard application of Theorem 1.4 resembling the application of normal coordinates in Riemannian geometry.

Theorem 4.1. Let

P (Γ_{ijk}(x), \frac{∂}{∂x^{l}} Γ_{ijk}(x), \frac{∂^{2}}{∂x^{l_{1}}∂x^{l_{2}}} Γ_{ijk}(x), …)

be a polynomial with constant coefficients in Christoffel symbols and their partial derivatives for some Darboux local coordinates. If this polynomial does not depend on the choice of Darboux coordinates, then there exists a polynomial

Q(R_{ijkl}, ∂_{m}R_{ijkl}, ∂_{m_{1}}∂_{m_{2}}R_{ijkl})

in the curvature tensor and its covariant derivatives such that P ≡ Q.

Proof. We use the fact that the value of P is independent of the choice of Darboux local coordinates. So, to calculate the value of P at a point x_{0} ∈ M, introduce week normal Darboux coordinates of Theorem 1.4 with the origin x_{0}. Thus, it is sufficient to prove that the partial derivatives of the Christoffel symbols at the origin are polynomials in the curvature tensor and its covariant derivatives. Let us denote

Γ_{ijk,l_{1}l_{2}...l_{m}} = \frac{∂^{m}}{∂x^{l_{1}} … ∂x^{l_{m}}} Γ_{ijk}(0). (4.1)

The property (1.3) means that the completely symmetric part Γ_{(ijk,l_{1}l_{2}...l_{m})} of the tensor (4.1) vanishes. On the other hand, there is a separate symmetry in i, j, k and l_{1}, l_{2}, …, l_{m}. Hence we may write (4.1) in the form

\frac{1}{(m + 3)!} ∑_{s∈S_{m+3}}(Γ_{ijk,l_{1}l_{2}...l_{m}} − Γ_{s(ijk,l_{1}l_{2}...l_{m})})

where S_{m+3} means the permutation group and s(ijk, l_{1}l_{2} … l_{m}) means the result of a permutation s applied to the whole set of indices. Since any permutation is a sequence of transpositions, each summand may be represented as a sum of the following terms

Γ_{ijk,l_{1}l_{2}...l_{m}} − Γ_{ijl,kl_{2}...l_{m}}

= \frac{∂^{m−1}}{∂x^{l_{2}} … ∂x^{l_{m}}} (\frac{∂Γ_{ijk}}{∂x^{l_{1}}} − \frac{∂Γ_{ijl_{1}}}{∂x_{k}} ) (0)

= −\frac{∂^{m−1}}{∂x^{l_{2}} … ∂x^{l_{m}}} R_{ijkl_{1}}(0) + \frac{∂^{m−1}}{∂x^{l_{2}} … ∂x^{l_{m}}} (Γ_{ikp}Γ_{jl_{1}q} − Γ_{il_{1}p}Γ_{jkq})(0)ω^{pq}

= −∂_{l_{2}} … ∂_{l_{m}}R_{ijkl_{1}}(0) + P_{m−1},

where P_{m−1} denotes a polynomial in partial derivatives of Γ_{ijk} at the origin containing the derivatives of the order at most m − 1. Thus, the theorem follows by induction.

5. Counterexample

The aim of this section is to show that the problem of finding Darboux coordinates is a non-correct one. The term correctness means as usual the following three properties: existence, uniqueness and smooth dependence on parameters. We construct a counterexample showing that at least the third property is violated. Of course, it would be more interesting to prove directly that Equation (3.5) has no smooth solutions with a third-order zero at the origin, but we managed to prove it only for variational equation. The reason is that the linearized equation is easier to treat than the original non-linear one.

To derive the linearization of (3.5), consider Christoffel symbols in the form

Γ_{ijk}(x) + τδΓ_{ijk}(x) (5.1)

where Γ_{ijk} satisfies (1.3), δΓ_{ijk} is an arbitrary symmetric tensor and τ is a small parameter. At τ = 0 we have a trivial solution S = 0 of (3.5) and we look for a variation of the trivial solution. Thus, we set for τ ≠ 0

S = τδS(z) + o(τ). (5.2)

Substituting (5.1) and (5.2) into (3.5) and extracting linear terms in τ, we come to the equation

2\frac{∂^{3}δS(z)}{∂z^{i}∂z^{j}∂z^{k}} z^{i}z^{j}z^{k} + 6Γ_{ijk}(z)z^{j}z^{k}∇^{i} (z^{l} \frac{∂δS}{∂z^{l}} − 2δS)

= −δΓ_{ijk}(z)z^{i}z^{j}z^{k}, (5.3)

the variation of (3.5). Introducing new functions

u = z^{l} \frac{∂δS}{∂z^{l}} − 2δS,

f = −\frac{1}{2}δΓ_{ijk}(z)z^{i}z^{j}z^{k},

rewrite (5.3) in the form

x^{i}x^{j} \frac{∂^{2}u}{∂x^{i}∂x^{j}} + 3Γ_{ijk}(z)z^{i}z^{k}∇^{i}u = f.

Further, differentiating the identity Γ_{jkl}(z)z^{j}z^{k}z^{l} ≡ 0, we get

q_{i}(z) := 3Γ_{ijk}(z)z^{j}z^{k} = −\frac{∂Γ_{jkl}(z)}{∂z^{i}}z^{j}z^{k}z^{l},

hence our equation becomes

z^{i}z^{j} \frac{∂^{2}u}{∂z^{i}∂z^{j}} + q_{i}(z)∇^{i}u = f(z) (5.4)

where f(z) may be an arbitrary function having a third-order zero at the origin. The coefficients q_{j}(z) have also a third-order zero and satisfy the relation

z^{i}q_{i}(z) = z^{i} \frac{∂}{∂z^{i}} (Γ_{jkl}(z)z^{j}z^{k}z^{l}) − 3Γ_{jkl}(z)z^{j}z^{k}z^{l} ≡ 0. (5.5)

Equation (5.4) exhibits quite different properties depending on dim M. First we consider briefly the case dim M = 2. The relation (5.5) takes the form

z^{1}q_{1}(z) + z^{2}q_{2}(z) = 0,

so that

q_{1} = a(z)z^{2}; q_{2} = −a(z)z^{1}

where

a(z) = ∫_{0}^{1} \frac{∂q_{1}}{∂z^{2}} (z^{1}, tz^{2})dt = − ∫_{0}^{1} \frac{∂q_{2}}{∂z^{1}} (tz^{1}, z^{2})dt.

Thus, a(z) is a smooth function having at least a second-order zero at the origin. Further, taking ω^{12} = −ω^{21} = 1, we get

q_{i}∇^{i}u = q_{1}ω^{12} \frac{∂u}{∂z^{2}} + q_{2}ω^{21} \frac{∂u}{∂z^{1}}

= a(z) (z^{1} \frac{∂u}{∂z^{1}} + z^{2} \frac{∂u}{∂z^{2}} ) ,

so that (5.4) may be rewritten in the form

z^{i}z^{j} \frac{∂^{2}u}{∂z^{i}∂z^{j}} + a(z)z^{i} \frac{∂u}{∂z^{i}} = f(z). (5.6)

Restricting this equation to the rays, we come to an ordinary differential equation

\frac{d^{2}u(tz)}{dt^{2}} + \frac{a(tz)}{t} \frac{du(tz)}{dt} = \frac{f(tz)}{t^{2}}.

It may be solved explicitly giving a unique solution of (5.6) with a third-order zero

u(z) = ∫_{0}^{1} dt ∫_{0}^{t} \frac{f(τz)}{τ^{2}} exp (− ∫_{t}^{t} au\frac{a(sz)}{s}ds) dτ.

In contrast, for dim M > 2, Equation (5.4) can not be reduced to an ordinary differential equation along the rays and this is a reason why the existence theorem is not true. We construct a counterexample for dim M = 4 taking non-vanishing components ω^{ij} in the form ω^{12} = −ω^{21} = 1; ω^{34} = −ω^{43} = 1 and defining non-vanishing components of Γ_{ijk}(z) for i ≤ j ≤ k by

Γ_{113} = Γ_{223} = \frac{1}{3}z^{3}, Γ_{114} = Γ_{224} = \frac{1}{3}z^{4},

Γ_{113} = Γ_{144} = −\frac{1}{3}z^{1}, Γ_{233} = Γ_{244} = −\frac{1}{3}z^{2}.

The coefficients q_{i}(z) are then equal to

q_{1}(z) = z_{1}(z_{3}^{2} + z_{4}^{2}), q_{2}(z) = z_{2}(z_{3}^{2} + z_{4}^{2})
q_{3}(z) = −z_{3}(z_{1}^{2} + z_{2}^{2}), q_{4}(z) = −z_{4}(z_{1}^{2} + z_{2}^{2}),

so that (5.5) is obviously fulfilled (here we have used subscripts instead of superscripts not to confuse them with squares). Thus,

q_{i}∇^{i}u = q_{1} \frac{∂u}{∂z_{2}} − q_{2} \frac{∂u}{∂z_{1}} + q_{3} \frac{∂u}{∂z_{4}} − q_{4} \frac{∂u}{∂z_{3}}

= (z_{3}^{2} + z_{4}^{2}) (z_{1} \frac{∂u}{∂z_{2}} − z_{2} \frac{∂u}{∂z_{1}} )

−(z_{1}^{2} + z_{2}^{2}) (z_{3} \frac{∂u}{∂z_{4}} − z_{4} \frac{∂u}{∂z_{3}} )

= r_{2}^{2} \frac{∂u}{∂φ_{1}} − r_{1}^{2} \frac{∂u}{∂φ_{2}}

where in the last line we have introduced polar coordinates r_{1}, φ_{1} in the plane z_{1}, z_{2} and r_{2}, φ_{2} in the plane z_{3}, z_{4}. In these polar coordinates Equation (5.4) takes the form

r_{i}r_{j} \frac{∂^{2}u}{∂r_{i}∂r_{j}} + r^{2} \frac{∂u}{∂φ_{1}} − r_{1}^{2} \frac{∂u}{∂φ_{2}} = f (5.7)

(summation over i, j = 1, 2 is understood).

Suppose that f is a sufficiently smooth function and u is a classical solution of (5.7) satisfying u = O(|z|^{3}). Denoting u_{m,n}(r_{1}, r_{2}), f_{m,n}(r_{1}, r_{2}) Fourier coefficients, we obtain from (5.7)

r_{i}r_{j} \frac{∂^{2}u_{mn}}{∂r_{i}∂r_{j}} + (imr_{2}^{2} − inr_{1}^{2})u_{mn} = f_{mn}.

Now the restriction to the ray is possible giving for the function u_{mn}(tr_{1}, tr_{2}) an ordinary differential equation

\frac{d^{2}u_{mn}(tr_{1}, tr_{2})}{dt^{2}} + (imr_{2}^{2} − inr_{1}^{2})u_{mn}(tr_{1}, tr_{2})

= \frac{1}{t^{2}} f_{mn}(tr_{1}, tr_{2})

which can be solved explicitly. Here t ∈ [0, ∞) and the right-hand side satisfies an estimate O(t). Taking into account that u_{mn} = O(t^{3}), we obtain that u_{mn} is uniquely defined by the following formula

u_{mn} = ∫_{0}^{t} \frac{sinh \sqrt{inr_{1}^{2} − imr_{2}^{2}}(t − τ)}{\sqrt{inr_{1}^{2} − imr_{2}^{2}}} \frac{1}{τ^{2}} f_{mn}(τr_{1}, τr_{2})dt. (5.8)

Let us take, in particular,

f = r_{1}^{2}r_{2}^{2k} ∑_{n=−∞}^{∞} \frac{e^{inφ_{2}}}{n^{2k}} (5.9)

where k ∈ ℕ. Clearly, this function belongs to C^{2k−2}(ℝ^{4}). From (5.8) it follows that only u_{0n} do not vanish. Denoting them simply u_{n} and taking t = 1 in (5.8), we obtain

u_{n}(r_{1}, r_{2}) = \frac{r_{1}^{2}r_{2}^{2k}}{n^{2k}} ∫_{0}^{1} \frac{sinh \sqrt{in}r_{1}(1 − τ)}{\sqrt{in}r_{1}}τ^{2k}dτ

= \frac{(2k)!}{(inr_{1})^{k+1}} (cosh \sqrt{in}r_{1} − ∑_{l=0}^{k} \frac{(inr_{1}^{2})^{l}}{(2l)!} ) \frac{r_{1}^{2}r_{2}^{2k}}{n^{2k}} .

Thus, if there exists a classical solution of (5.7), it should be represented by the Fourier series

(2k)!r_{1}^{2}r_{2}^{2k} ( ∑_{n=−∞}^{∞} \frac{cosh \sqrt{in}r_{1}}{(inr_{1})^{k+1}n^{2k}} e^{inφ_{2}}

− ∑_{n=−∞}^{∞} ∑_{l=0}^{k} \frac{e^{inφ_{2}}}{(inr_{1})^{k−l+1}(2l)!n^{2k}} ) , (5.10)

and this series must be convergent since the classical solution belongs at least to C^{2}. But the series (5.10) is divergent for any r_{1} ≠ 0. Indeed, the second series in (5.10) converges, while the first one has coefficients proportional to

\frac{cosh \sqrt{in}r_{1}}{n^{3k+1}} ,

so they do not tend to 0 for n → ±∞.

So, the classical solution of (5.7) does not exist in any neighborhood of the origin although the right-hand sides (5.9) may be of arbitrary high smoothness.

References

[GRS] I. Gelfand, V. Retakh and M. Shubin, Fedosov Manifolds Advances in Mathematics, 136 (1998), 104–140.

[F] B. Fedosov, Deformation Quantization and Index Theory, Akademie Verlag, Berlin, 1996.

[NN] A. Newlander and L. Nirenberg, Complex Analytic Coordinates in Almost Complex Manifolds, Annals of Math., 65 1957, 391–404.

[W] H. Weyl The Classical Groups, Princeton University Press, Princeton, New Jersey, 1939.

Mathématique Physique, Université de Bourgogne, BP 47870, F-21078, Dijon Cedex, France

    E-mail address: fedosov@u-bourgogne.fr