On Weyl’s type theorems and genericity of projective rigidity in subRiemannian Geometry
Abstract.
H. Weyl in 1921 ([8]) demonstrated that for a connected manifold of dimension greater than , if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one. In the present paper we investigate the analogous property for subRiemannian metrics. In particular, we prove that the analogous statement, called the Weyl projective rigidity, holds either in real analytic category for all subRiemannian metrics on distributions with a specific property of their complex abnormal extremals, called minimal order, or in smooth category for all distributions such that all complex abnormal extremals of their nilpotent approximations are of minimal order. This also shows, in real analytic category, the genericity of distributions for which all subRiemannian metrics are Weyl projectively rigid and genericity of Weyl projectively rigid subRiemannian metrics on a given bracket generating distributions. Finally, this allows us to get analogous genericity results for projective rigidity of subRiemannian metrics, i.e. when the only subRiemannian metric having the same subRiemannian geodesics , up to a reparametrization, with a given one, is a constant scaling of this given one. This is the improvement of our results on the genericity of weaker rigidity properties proved in recent paper [5].
Key words and phrases:
SubRiemannian geometry, Riemannian geometry, Conformal geometry, Projective Geometry, Normal Geodesics, Abnormal Geodesics, Nilpotent Approximation2010 Mathematics Subject Classification:
53C17, 53A20, 53A301. Statement of the problem and main results
In Riemannian geometry, projectively (or geodesically) equivalent metrics are Riemannian metrics on the same manifold which have the same geodesics, up to reparameterization. The local classification of all pairs of projectively equivalent Riemannian metrics under natural regularity assumptions was maid by LeviCivita in 1896 ([7]). This paper is devoted to the projective equivalence of more general class of metrics, the subRiemannian metrics and is a continuation of our recent work [5].
A subRiemannian manifold is a triple , where is a smooth connected manifold, is a distribution on (i.e. a subbundle of ) which is assumed to be bracket generating everywhere in the sequel without special mentioning, and is a Riemannian metric on , and thus defines an Euclidean structure on every fiber of . We say that is a subRiemannian metric on . Consider the optimal control problem of minimizing the corresponding energy functional on the space of absolutely continuous curve tangent to . The geodesics of the subRiemannian metric are projections of the Pontryagin extremals for this problem. SubRiemannian Pontryagin extremals and the corresponding geodesics can be of two types, normal or abnormal.
The normal Pontryagin extremal of the subRiemannian metric are integral curves of the Hamiltonian system for the corresponding Hamiltonian , living on a nonzero level set of this Hamiltonian. The Hamiltonian of the subRiemannian metric is defined by
(1.1) 
where
The abnormal Pontryagin extremals live in the zero level set of , or, equivalent, on the annihilator of the distribution , i.e.
(1.2) 
Their description is more involved, as they are not the integral curve of the subRiemannian Hamiltonian , and will be given in section 5. Abnormal geodesics depend only on the distribution , not on , so they are automatically the same for all subRiemannian metrics on the same distribution.
Riemannian metrics appear as the particular case of subRiemannian ones, where . The classical Riemannian geodesics can be equivalently described as the normal geodesics coming from the corresponding Hamiltonian (1.1). Riemannian metrics do not have abnormal geodesics. We thus extend the definition of projectively equivalence to subRiemannian metrics in the following way.
Definition 1.1.
Let be a manifold and be a bracket generating distribution on . Two subRiemannian metrics and on are called projectively equivalent at if they have the same geodesics, up to a reparameterization, in a neighborhood of .
The trivial example of projectively equivalent metrics is the one of two constantly proportional metrics and , where is a real number. We thus say that these metrics are trivially (projectively or affinely) equivalent.
Definition 1.2.
A subRiemannian metric on is said to be projectively rigid if it admits no nontrivially projectively equivalent metric.
It is still a widely open problem to classify all pairs of projectively equivalent subRiemannian metrics. A much easier task is to study whether the projectively rigid subRiemannian metrics form a generic set in the space of all subRiemannian metrics on a connected manifolds. In studying this question one naturally arrives to the following weaker (intermediate) notion of rigidity.
Definition 1.3.
A subRiemannian metric is said to be conformally projectively rigid if any metric projectively equivalent to is conformal to .
In our recent paper [5] we proved the following genericity results for conformally projective rigidity:
Theorem 1.4.
Let be a smooth manifold and be a smooth distribution on . A generic subRiemannian metric on is conformally projectively rigid.
Theorem 1.5.
Let and be two integers such that , and assume and if is even. Then, given a smooth dimensional manifold and a generic smooth rank distribution on , any subRiemannian metric on is conformally projectively rigid.
The latter theorem is based on the following result also proved in [5].
Theorem 1.6.
If is a bracket generating distribution on a connected manifold such that the nilpotent approximation of it at every point of an open and dense subset of does not admit a product structure, then any subRiemannian metric on is conformally projectively rigid.
In light of these results, it is natural to ask whether conformally projective rigidity can be replaced by just projective rigidity in both of these theorems. In the Riemannian case H. Weyl in 1921 ([8]) demonstrated that for , if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one.^{1}^{1}1In fact this is a simple consequence of the LeviCivita classification in [7] which was written much earlier than [8] but we prefer to relate it first to H. Weyl as the great founder of both conformal and projective geometry.
Definition 1.7.
A metric is said to be Weyl projectively rigid if any metric, which is simultaneously conformal to and projectively equivalent to is constantly proportional to .
So, in this terminology the Weyl theorem says that for any Riemannian metric is Weyl projectively rigid. While in Riemannian case the proof of this result is rather trivial, it is not known yet whether the same statement is true for all subRiemannian metrics. The main problem that arises in trying to prove this statement for the general subRiemannian case is the presence of abnormal extremals. The main objective of the present paper is to study under what conditions are subRiemannian metrics Weyl projectively rigid. Studying the solvability of the equations for projective equivalence, one inevitably arrives to the questions of divisibility of certain polynomials on the fibers of the cotangent bundle , so it is natural to complexify the picture by complexifying not only the fibers of but the manifold itself which is possible, at least locally, under assumption that is real analytic. This leads naturally to the necessity to consider the notion of complex extremals and geodesics. Very roughly speaking, we show that a real analytic subRiemannian metric is Weyl projectively rigid either if the underlying distribution does not have too much complex abnormal geodesics through a point or does not have too much complex nonstrictly normal geodesics, i.e. complex normal geodesics which are simultaneously abnormal. Our condition are already enough to prove the Weyl projective rigidity for appropriate generic class of subRiemannian metrics in real analytic category (see Theorems 1.12 and 1.18) below) so that it is possible to replace conformally projective rigidity by projectively rigid in Theorems 1.4 and 1.5.
Now we will describe our results in more detail. Assume that is a real analytic subRiemannian structure. Locally (i.e. in a neighborhood of any points of ) we can consider a complex manifold , a complexification of , by extending the transition maps between charts, which are real analytic by definition, to analytic functions. We can extend locally the (realanalytic) distribution and subRiemannian metric to the (complex) analytic distribution and a field of symmetric forms on each fiber of this distribution.
We can also consider the (complex) cotangent bundle of whose fibers are complex vector spaces. We can extend the subRiemannian Hamiltonian defined by (1.1) analytically to the complex Hamiltonian on this bundle and consider the corresponding complex Hamiltonian vector field. The complex normal extremals are by definition the integral curves of this vector field and the complex normal geodesics are projections of these integral curves to .
Remark 1.8.
Note that after the complexification the zerolevel set of the complex subRiemannian Hamiltonian is strictly larger that the annihilator . The integral curves of the complexified Hamiltonian lying in play the same role as null geodesics in the pseudoRiemannian geometry and, in particular, they are the same, up to reparameterization, for all subRiemannian metrics from the same conformal class. We will call them and their projections to the complex null normal extremals and geodesics, respectively.
Also, we can define Jacobi curve and the corresponding osculating flag for every complex normal extremal. Further, manipulating with the annihilators of the complex distribution in the complex cotangent bundle of , similarly to the standard real case, we can define complex abnormal, and consequently the strictly normal subRiemannian geodesics (see section 5 for more detail).
We also need the notion of a corank of a geodesics. From now on by dimensions we will mean complex dimensions. Given a complex normal geodesic of a subRiemannian metric we say that a complex Pontryagin normal extremal projected to as parameterized curve is a Pontryagin normal lift of . Given an abnormal geodesic of a distribution we say that a complex Pontryagin abnormal extremal projected to is a Pontryagin abnormal lift of . The projection of this lift to the projectivized cotangent bundle will be called the projectivized Pontryagin abnormal lift of . The corank of a complex normal geodesics of a subRiemannian metric is by definition the dimension of the affine space of the normal Pontryagin lifts of . The corank of a normal geodesic is a nonnegative integer.
The corank of a complex abnormal geodesic of a distribution is by definition the dimension of the (vector) space of all its abnormal lifts. The corank of an abnormal geodesic is a positive integer.
Remark 1.9.
Note that if and are two distinct normal lifts of a normal geodesics (which are either both null or both nonnull), then is an abnormal lift of . Similarly, if is an abnormal lift of a geodesic and is a normal lift of , then is a normal lift of .
From the previous remark it follows that a normal geodesic is simultaneously an abnormal geodesic if and only if its corank is greater than and in this case the corank of as normal geodesics is equal to the corank of as abnormal geodesics.
Further, if given positive integers and , there exists a nonzero number and a dimensional submanifold of the level set of the complexified subRiemannian Hamiltonian which is foliated by complex normal extremals of corank , then we say that the projections of these extremals to form an parametric family of complex normal geodesics of corank .
Also note that the nilpotent approximation (even of a smooth but not real analytic subRiemannian structure) is always real analytic as it is a leftinvariant structure on a nilpotent Lie group. The following two theorems are our main results on Weyl projective rigidity in terms of normal geodesics:
Theorem 1.10.
Assume that is a smooth subRiemannian manifold such that its nilpotent approximation at every point of an open and dense subset of satisfies the following property: for every positive , there is no parametric family of corank nonstrictly normal complex geodesics through a point. Then the metric is Weyl projectively rigid.
Theorem 1.11.
Assume that is a real analytic subRiemannian manifold such that there is no open set in with the following property: for some positive integer through any point there is an parametric family of corank nonstrictly normal complex geodesics. Then the metric is Weyl projectively rigid.
These two theorems will be proved in section 4, based on more general but technical Theorem 3.2, proved in section 3.
The conclusion of Theorems 1.10 and 1.11 for a subRiemannian structure holds in particular when either the subRiemannian structure is smooth, and its nilpotent approximation do not have complex nonstrictly normal geodesics or the subRiemannian structure is real analytic and does not have complex nonstrictly normal geodesics. The latter holds generically. It follows from the complex analog of [3, Proposition 2.22], which has literally the same proof. To summarize, we have the following result on genericity of the Weyl rigidity:
Corollary 1.12.
Let be a real analytic manifold and be a real analytic distribution on of rank greater than . A generic real analytic subRiemannian metric on is Weyl rigid.
Now we formulate our main results on Weyl projective rigidity in terms of complex abnormal extremals. We will use the notion of an abnormal extremal of minimal order, introduced in [2], see Definition 5.2 below. The condition of minimal order implies in particular that on a set of full measure such abnormal extremal is tangent to a prescribed line or equivalently, on a set of full measure the germ of the extremal at any point of this set is uniquely determined by this point. The following two theorems will be proved in section 5.
Theorem 1.13.
Assume that is a smooth distribution on a connected manifold such that its nilpotent approximation at every point of an open and dense subset of satisfies the following properties: every complex abnormal extremal of the nilpotent approximation is of minimal order. Then any smooth subRiemannian metric on is Weyl projectively rigid.
Theorem 1.14.
Assume that is a real analytic distribution on a connected manifold such that every complex abnormal extremal of is of minimal order. Then any real analytic subRiemannian metric on is Weyl projectively rigid.
Corollary 1.15.
If is a smooth bracket generating distribution on a connected manifold such that the nilpotent approximation of it at every point of an open and dense subset of does not admit a product structure and every complex abnormal extremal of is of minimal order, then any smooth subRiemannian metric on is projectively rigid.
Based on the last corollary we can easily find many new classes of distributions on connected manifolds for which all subRiemannian metrics on them are projectively rigid (before this statement was known for contact distributions only ([9])). For example, this will be true for the following distributions, for which it is easy to see that all possible nilpotent approximations satisfy conditions of Corollary 1.15:

Engel distributions, i.e. rank distributions on dimensional manifolds with the small growth vector ;

Rank distributions on dimensional manifolds with the small growth vectors ;

rank distributions on dimensional and dimensional manifolds with the small growth vectors and , respectively;

Rank distributions on dimensional manifolds with the small growth vectors ;

Rank distributions on dimensional manifolds with the small growth vectors or .
Remark 1.16.
Further, the main result of [2, Theorem 2.4] states that all (real) abnormal extremals of a generic smooth rank distribution are of minimal order and corank . This result can be literally extended to the complex abnormal exttremals of real analytic manifolds), because the genericity condition in [2, Theorem 2.4] is given by the complement of algebraic conditions with respect to the fibers of so that the complexification can be done. So, the following theorem holds.
Theorem 1.17.
All complex abnormal extremals of a generic rank real analytic distribution distribution are of minimal order and corank .
Combining this theorem with Theorem 1.13 we get one more genericity results on the Weyl rigidity.
Corollary 1.18.
Let and be two integers such that . On a generic real analytic rank distribution on a connected dimensional real analytic manifold any subRiemannian metric is Weyl projectively rigid.
Finally, as immediate consequences of Theorems 1.4 and 1.12 and Theorems 1.5 and 1.18, respectively, we get the following two genericity results for projective rigidity, improving the main results of [5].
Corollary 1.19.
Let be a real analytic manifold and be a distribution on . A generic real analytic subRiemannian metric on is projectively rigid.
Corollary 1.20.
Let and be two integers such that , and assume and if is even. Then, given a smooth dimensional manifold and a generic smooth rank distribution on , any subRiemannian metric on is projectively rigid.
2. The fundamental algebraic system in the conformal case
2.1. Equations for orbital diffeomorphisms in local coordinates
In this subsection, following [5], we introduce orbital diffeomorphisms between extremal flows and explain their relation to the projective equivalence, then deduce the equations for orbital diffeomorphisms in local metric for projective equivalent and conformal subRiemannian metric. All formulas here can be directly derived from the corresponding formulas in [9] and [5], where the general case (of not necessary conformal but projective equivalent subRiemannian metrics) is considered. To make the presentation selfcontained we derive all formulas here in the particular conformal case.
Let be a manifold and be a bracket generating distribution on . We consider two subRiemannian metrics on that are both conformal and projectively equivalent. Let us denote these metrics by and , where is a never vanishing smooth function. Let and be the subRiemannian Hamiltonians of and , respectively. Obviously
(2.1) 
Denote and the respective
One says that and are orbitally diffeomorphic on an open subset of if there exists an open subset of and a diffeomorphism such that is fiberpreserving, i.e. , and sends the integral curves of to the reparameterized integral curves of , i.e., there exists a smooth function with such that for all and for which is well defined. Equivalently, there exists a smooth function such that
(2.2) 
The map can be extended as a mapping from to itself by rescaling, i.e.,
The resulting map is called an orbital diffeomorphism between the extremal flows of and . In the considered case, from (2.1) and the fact that is fiberpreserving it follows immediately that the function in (2.2) coincides with the function , i.e. we have
(2.3) 
In [5] we established the relationship between projective equivalence of subRiemannian and orbital equivalence of the corresponding subRiemannian Hamiltonians. In particular, in Proposition 3.4 there we proved that there exists a local orbital diffeomorphisms between the Hamiltonian vector fields associated with and near generic^{2}^{2}2In fact in the original formulation in [5] we used the term ample instead of generic, see [Definition 2.9] there, but we do not really need this technicalities here. point of .
Now we will work in coordinates on fibers of induced by an appropriate local moving frame on . Fix a point and choose a frame of adapted to at such that is a orthonormal frame of . At any point in a neighborhood of , the basis of induces coordinates on defined as . These coordinates in turn induce a basis of for any . For , we define the lift of as the (local) vector field on such that and . In this way the local frame on induces the local frame
(2.4) 
on . By a standard calculation, we obtain and
(2.5) 
where , are the structure functions of the frame , defined near by
Further, from (2.1) it follows that
(2.6) 
Finally, let us denote by the component of on the fiber, i.e. . From (2.1) it follows that
(2.7) 
Substituting this into (2.3) we get the following:^{3}^{3}3From now on to simplify the notation in all relation involving functions on open sets of actually will mean .
Lemma 2.1.
The map is an orbital diffeomorphism between extremal flows of subRiemannian metrics and if and only if the components satisfy the following system of equations:
(2.8)  
(2.9)  
(2.10) 
where .
Equations (2.8)(2.9) are obtained by straightforward calculations in the moving frame (2.4) after plugging equations (2.5), (2.6), and (2.7) into (2.3). Equations (2.8) are obtained by comparison of the components of of both sides of (2.3) with , while equations (2.9) are obtained by comparison of the components of of both sides of (2.3) with .
2.2. Fundamental algebraic system
Now following [5] again we replace the system (2.8)(2.9) that contains derivatives of the unknown functions , by the (infinite) linear algebraic i.e. without derivatives) system for that unknown function that we call the fundamental algebraic system. The process of obtaining the latter can be seen in a sense as the infinite prolongation of the subsystem given by (2.8) using in each step of the prolongation the equations from (2.9).
In more details, in the first step one differentiate each of equations from (2.8) in the direction of and replace each in the resulting expression by the righthand side of (2.9). In this way we get new equations which are linear in . In the next step we differentiate these new equations in the direction of and replace each in the resulting expression by the righthand side of (2.9) to obtain new equations which are linear in . The fundamental algebraic system is obtained by repeating this process infinitely many times. Setting and , the fundamental algebraic system [5, (3.8)] writes as
(2.11) 
where is the matrix defined recursively in [5, (3.10)] and is a column vector with an infinite number of rows which can be decomposed in layers of rows as
(2.12) 
where the coefficients , , of the vector are defined by
(2.13) 
Note that by [5, Proposition 3.11] the matrix is injective at a generic .
2.3. Sufficient conditions for Weyl rigidity in terms of solutions of the fundamental algebraic system
The fundamental algebraic system (2.11) implies that the coordinates of are rational functions on the fibers. Proving that and are proportional actually amounts to prove that these coordinates are polynomial, as stated below.
Proposition 2.2.
If there exists a local orbital diffeomorphism which is polynomial on the fibers, then and are locally constantly proportional, i.e., is constant.
Before giving the proof of this result, we need to study the consequence of the fundamental algebraic system on the nilpotent approximation.
Fix a regular point and denote by the nilpotent approximation of at . We argue as in the proof of [5, Theorem 7.1], with the same notations. In particular is a frame of adapted to such that is orthonormal and is the matrix of [5, Proposition 3.10] constructed by using as a frame.
Lemma 2.3.
There exists one, and only one, solution s of
(2.14) 
where, for any and , is defined by
(2.15) 
Proof.
An easy induction argument based on equations (2.13) shows the following result, similar to [5, Lemma 7.4]: for any and , there hold:

for every near , is a polynomial in of weighted degree

the homogeneous term of highest weighted degree in is .
It results from (2.11) that is not of full rank, thus also the matrix is not of full rank. Since is of full rank at a generic by [5, Proposition 3.11], there exists a unique element in of the form , which ends the proof. ∎
Using all equations above it is easy to show that has the following properties.

[(i)]

Each , , is a rational function which is:

homogeneous of degree w.r.t. the usual degree;

homogeneous with .


For , we have
(2.16) where .

For , we have
(2.17)
Lemma 2.4.
Assume that the map given in Lemma 2.3 is polynomial. Then
(2.18) 
Proof.
By hypothesis, every , , is a polynomial. Moreover, by Property 1 above, is a linear function of and depends only on the coordinates of weight . To simplify the notations, we use the following convention: given a positive integer , an index denotes an index of weight and denotes . With this notation we have, for every ,
(2.19) 
where the coefficients are real numbers. Taking the derivative along we obtain
(2.20) 
On the other hand, plugging (2.19) into (2.17), we get
(2.21) 
Fix an index . By identifying the coefficients of the monomial in (2.20) and (2.21), we obtain the following equality,
(2.22) 
and, after a summation on the indices ,
(2.23) 
Set . Then the above equality writes as
(2.24) 
Note that since is the nilpotency step. Hence,
(2.25) 
Now, by plugging (2.19) in (2.16), we have, for :
(2.26) 
Given an index , the identification of coefficient of in this equality gives
(2.27) 
and by summation on the indices , we obtain
(2.28) 
This equation and (2.25) imply , which ends the proof. ∎
Proof of Proposition 2.2..
Assume to be defined on an open subset of . Fix a regular point in and let be the nilpotent approximation of at .
Let be a nonzero maximal minor of . It is a homogeneous polynomial which is the homogeneous part of highest weighted degree of the corresponding minor (same rows and columns) of , which is nonzero as well. It results easily from (2.11) that, for , we have where , and , where is the homogeneous part (eventually zero) of weighted degree in . From the hypothesis of the theorem, is polynomial, therefore is polynomial as well and by Lemma 2.4 we get , .
Since regular points form an open and dense subset of , the functions , , are identically zero on . The family being a Liebracket generating family, we thus obtain that is locally constant. ∎
2.4. A remark on Lemma 2.3
Let be the realvalued function on defined by
(2.29) 
In a system of privileged coordinates at such that , writes as
(2.30) 
The existence of the mapping in Lemma 2.3 may be interpreted as follows.
Lemma 2.5.
There exists a fiberpreserving map such that, on a neighbourhood of every ample covector (w.r.t. ), is smooth and sends the integral curves of the Hamiltonian vector fields of the metric to the ones of .
3. The most general subRiemannian Weyl type theorem
In this section we formulate the most general technical version of the subRiemannian Weyl theorem that we were able to obtain. The versions of the subRiemannian Weyl theorem (Theorems 1.10 and 1.11) formulated in the Introduction will follow from its proof.
We start with the following.
Definition 3.1.
Given an open subset of the function on the complexified cotangent bundle is called a polynomial with respect to the fibers over if, in the canonical coordinates induced by some local coordinates in , is represented as a polynomial with respect to the fibers with coefficients being holomorphic function of the base . Further, given a point a germ over of polynomials with respect to the fibers of cotangent bundle is an equivalence class of such polynomials so that two polynomials are equivalent if they coincide over a neighborhood of a point .
Theorem 3.2.
Assume that is a real analytic subRiemannian manifold such that there is no nonconstant polynomial with respect to the fibers of over some open set of , such that an open subset of the zerolevel set of is a manifold foliated by complex normal extremals each of which projects to nonstrictly normal geodesics. Then the subRiemannian metric is Weyl projectively rigid.
Proof.
Consider the map , which is a solution of (2.11). Then by Proposition 2.2, to get the conclusion of our theorem we only need to show that is polynomial with respect to the fibers near a point .
Given a positive integer denote by the truncation up to the th layer of the fundamental matrix from (2.11). By [5, Proposition 3.11] we can choose large enough so that at least one minor of size in is not identically zero.
From now on we work on the complexified manifold . The corresponding complexified cotangent bundle can be identified locally with (where is identified with ). Let be the set of germs of holomorphic functions on at . Under the above identification, a germ of polynomials with respect to the fibers of in the sense of Definition 3.1 can be seen as polynomials on (the second copy of) with coefficients in . Since is a factorial ring (or unique factorization domain), the set of these polynomials form a factorial ring as well (see for instance [6]), which mean that every element can be written as a product of irreducible elements, uniquely up to order and units.
It results from (2.11) that for any nonzero minor of we have
(3.1) 
where and are polynomial in . Canceling the greatest common factor of the collection of polynomials , we get a collection of polynomials with the greatest common factor equal to constant and such that
(3.2) 
Besides, substituting into (2.9) we get
(3.3) 
Let us show that under the assumption of Theorem 3.2 is constant. Assuming the converse, there is an irreducible polynomial in such that , where is a positive integer and is a polynomial such that and are coprime. By constructions, there exists such that is not divisible by , otherwise is a nonconstant common factor of the collection .
Consider this particular . Although the polynomials and are not coprime in general, if we further reduce the expression (3.2) for to the lowest terms (i.e. such that the numerator and denominator will be coprime), then the denominator will be divisible by . Note that in (3.1) we can use any nonzero maximal minor of and the expression for in the lowest terms is unique and does not depend on the initial choice of the nonzero maximal minor . Hence, is a common divisor of all maximal minors of .
From (3.3), there holds
which implies that is divisible by , and so that
Since and are not divisible by the irreducible polynomial , we conclude that is divisible by .
Denote by the zerolevel set of (it is an analytic subset of near the fiber ). We have shown that is zero on . Note that, although is irreducible over , its restriction to some fiber might be reducible over . However, the restriction on the generic fiber (in the domain of definition of ) is irreducible. This implies that is not identically zero on . Indeed, assume the converse. Since is not constant, there exists such that is not zero but on . By applying the Hilbert Nullstellensatz to each generic fiber must belong to the radical of the ideal generated by the restriction of to the same fiber, but this is impossible as degree of the polynomial is smaller than degree of .
Denote by the subset of , where the vector ^{4}^{4}4Here we could impose a weaker condition but we will need the given stronger condition in the next section is not equal to zero. By constructions this is an open (and dense) subset of . Then, by constructions, is a submanifold of and is tangent to . Therefore, any complexified normal extremal of starting at a point of will stay in for sufficiently small time and is foliated by normal extremals.
Consider such a normal extremal , , in , so that . Given any define the filtration as follows. First, set to be the tangent space to the fiber of at , i.e.
where, as before, denotes the canonical projection. Note that defines a distribution on called the vertical distribution. Finally, define recursively
Since is a common divisor of all maximal minors of , due to [5, Lemma 3.12] there holds for all , and so
(3.4) 
(recall that from the beginning we have taken ). Further, for in an open and dense subset of the dimensions of the spaces