OVIDIU CALIN, DER-CHEN CHANG, AND JISHAN HU
Abstract: It is well known that the system ∂xf = a, ∂yf = b on R2 has a solution if and only if the closure condition ∂xb = ∂ya holds. In this case the solution f is the work done by the force U = (a, b) from the origin to the point (x, y).
This paper deals with a similar problem, where the vector fields ∂x, ∂y are replaced by the Heisenberg vector fields X1, X2. In this case the sub-Riemannian system X1f = a, X2f = b has a solution f if and only if the following integrability conditions hold X12b = (X1X2 + [X1, X2])a, X22a = (X2X1 + [X2, X1])b. The question addressed by this paper is whether we can provide a Poincar´e-type Lemma for the Heisenberg distribution. The positive answer is given by Theorem 2, which provides a result similar to the Poincar´e’s Lemma in the integral form. The solution f in this case is the work done by the force vector field aX1+ bX2 along any horizontal curve from the origin to the current point.
Key words: Heisenberg distribution, work, Poincar´e lemma MS Classification (2000): Primary: 93B05; Secondary: 49N99
1. Introduction
There are several equivalent versions of Poincar´e’s lemma. Some are stated in terms of exactness and closeness of 1-forms, others use conser-vative and gradient vector fields. The differential formulation can be found for instance in Spivak [8], while an elementary vectorial construction can be found in Yap [10]. For the sake of completeness we present next a brief review regarding a couple of elementary versions of Poincar´e’s lemma in di-mensions 2 and 3. These will be useful in understanding of what we are trying to do in the Heisenberg case. We recall that in two dimensions the result takes the following well-known form:
Let U be an open and contractible set in R2. Then, for a pair of smooth functions a and b
∂xb = ∂ya ⇐⇒
∃ a smooth function f
such that ∂xf = a and ∂yf = b.
The integrability condition on the left states that the vector field V = (a, b) is irrotational, while the condition on the right represents the fact that V is a gradient vector field, i.e., V = ∇f , or, equivalently, V is a conservative vector field.
This research project is partially supported by Hong Kong RGC-GRF research grants #601813 and #601410.
The equivalent formulation using 1-forms states that the 1-form ω = a(x, y)dx+ b(x, y)dy is exact, i.e., there is a smooth function f such that ω = df , if and only if ω is closed, i.e., dω = 0, a condition equivalent to ∂xb = ∂ya. It is worth noting that the solution f can be constructed explicitly and has a remarkable physical significance. Let r(t) = t(x, y) = (tx, ty) = x(t), y(t), 0 ≤ t ≤ 1, be the straight line segment from the origin to the point (x, y). Consider f to be the work done by the 1-form ω = adx + bdy from the origin to the point (x, y) along the curve r(t)
f (x, y) = Z r(t) ω = Z 1 0 (a, b) · ˙r(t) dt = Z 1 0
a(tx, ty)x + b(tx, ty)y dt. An elementary computation provides
∂xf (x, y) = a(x, y) + Z 1 0 ty∂b ∂x− ∂a ∂y dt (1.1) ∂yf (x, y) = b(x, y) + Z 1 0 tx ∂a ∂y − ∂b ∂x dt. (1.2) Consequently, if ∂a ∂y = ∂b ∂x,
then ∂xf = a, ∂yf = b. The converse also holds true and follows as a direct consequence of the symmetry of the second derivative.
It is worthy to rewrite equations (1.1)-(1.2) in the vector format (1.3) ∇f (x, y) = (a, b) + Z 1 0 ∂b ∂x − ∂a ∂y (ty, −tx) dt.
The 3-dimensional vector version of the Poincar´e lemma takes the follow-ing form. Let V be a smooth vector field on R3. Then
(1.4) V (r) = ∇ Z 1 0 V (tr) · r dt+ Z 1 0 curl V (tr) × tr dt, where r = (x, y, z) is the notation for points in R3. If we denote by
(1.5) f (r) =
Z 1
0
V (tr) · r dt
the work done by a force expressed by the vector field V along the curve {tr}, 0 ≤ t ≤ 1, then (1.4) takes the equivalent form
(1.6) ∇f (r) = V (r) +
Z 1
0
tr × curl V (tr) dt, Consequently, if curl V = 0, or, equivalently, if
∂V2 ∂x1 −∂V 1 ∂x2 = 0, ∂V 3 ∂x1 −∂V 1 ∂x3 = 0, ∂V 2 ∂x3 − ∂V 3 ∂x2 = 0, then the work given by (1.5) satisfies
∇f (r) = V (r).
The present paper provides a variant of Poincar´e’s lemma in the variant of formulas (1.1)–(1.2) for the case of the Heisenberg vector fields. This is a continuation of work done in articles [2] and [3]. The point of this paper
is to advance an explicit integral formula for the Heisenberg case, which is not easy to obtain in general on sub-Riemannian manifolds.
Why would a sub-Riemannian version of Poincar´e’s lemma be worthy to investigate? There are several good reasons for doing this.
First, it provides a necessary and sufficient condition for a sub-elliptic system of equations to have solutions. The sub-elliptic systems are systems of linear differential equations with the number of equations smaller than the dimension of the space; section 2 introduces these systems given by (2.1). The case when the number of equations equals the dimension of the space defines an elliptic system and the existence of its solutions is provided by the classical Poincar´e lemma. It is our hope that a Poincar´e lemma can become in the sub-Riemannian context as powerful and influential as its elliptical, classical, version.
Second, it states the relation with the non-holonomic mechanics, since the solution of a sub-elliptical system is the work done along a horizontal curve, i.e. a curve tangent to a given distribution. For instance, the work done by moving a ball between two given positions is the work along a horizontal curve in a certain distribution of a sub-Riemannian manifold and might be related to the solution of the associated sub-elliptical system.
The plan of the paper follows. Section 2 deals with a bird’s eye overview of the sub-Riemannian version of the problem and sub-Riemannian systems of equations, reviewing the current literature. Equivalent formulations using horizontal 1-forms and horizontal vector fields are presented in section 3. Section 4 contains the main result, which is a Poincar´e-type formula on the Heisenberg group.
2. Sub-Riemannian Systems of Equations
Consider a set of smooth vector fields {X1, X2, · · · , Xm} on Rn, where m < n. The vector fields span at each point p ∈ Rn a plane Hp. The smooth assignment p → Hp is called the horizontal distribution induced by {Xj}.
Consider a metric g defined on the horizontal distribution H, g : H × H → F (M ), with respect to which the vector fields {Xj} are orthonormal, i.e. g(Xi, Xj) = δij. We note that the extension of g to Rn is not unique, but this does not affect the study of sub-Riemannian geometry. The metric g can be used to measure lengths of horizontal vectors and horizontal curves, and it is called the sub-Riemannian metric. Here the set of smooth functions on Rn is denoted by F (Rn).
The pair (H, g) provides a sub-Riemannian structure on the space Rn. We shall refer to the vector fields contained in the distribution H as horizontal vector fields. The set of all horizontal vector fields is denoted by Γ(H).
Since the vector fields Xj can be viewed as the dual of m 1-forms {ω1, · · · , ωm} on Rn, one can deal with this geometry equivalently from the point of view of constraints. This approach was initiated by Vranceanu [9] in 1926, under the name of non-holonomic geometry.
A smooth curve in Rn is tangent to the distribution H if its velocity be-longs to the plane of the distribution at each point along the curve. A curve
of this type is called a horizontal curve. The thermodynamic interpretation of horizontal curves is that of the adiabatic processes; they are curves along which the exchange of heat is zero, see [1], p. 74. Adiabatic processes are important components of Carnot’s proposed engine in 1820s. Carath´eodory provided in his 1919 paper a necessary condition for the Pfaff system as-sociated with a horizontal distribution to admit connectivity by horizontal curves. This is the reason why sub-Riemannian geometry is also known under the name of Carnot-Carath´eodory geometry.
The following condition on H was used independently by Chow and Ra-shevskii ([5], [7]) to prove the global connectedness of Rn by horizontal curves. The same condition was also used by H¨ormander [6] as a suffi-cient, but not necessary condition for the differential operator P
jXj2 to be hypoelliptic.
The horizontal distribution H satisfies the bracket generating condition if the vector fields Xj, together with finitely many of their iterated brackets span the tangent space of Rnat each point. This means that for each p ∈ Rn, there is an r > 1 such that
Xi, · · · , [Xi, Xj], · · · , [Xi, [Xj, Xk]], · · · , [· · · [Xi1, · · · , [Xir, Xir+1]] · · · ]
span TpRn. If r = r(p) is the smallest integer with this property at the point p, then the number k(p) = r(p) + 1 is called the step of the distribution at the point p. Some authors use also the equivalent term of order of nonholon-omy. It is worth noting that k(p) depends on the point p and might exhibit discontinuities from point to point.
Recall now the open problem asked in [1], p. 51:
Given m smooth functions aj ∈ F (Rn), find a function f ∈ F (Rn) that satisfies the system
X1(f ) = a1 · · · · (2.1)
Xm(f ) = am.
Since the number of equations is smaller than the space dimension, m < n, the previous system is called sub-elliptic.
If H satisfies the bracket generating condition, then the solution f , if exists, is unique up to an additive constant. The regularity of the solution follows from H¨ormander’s theorem (see [6]) that yields the hypoellipticity of the operator P
jXj2.
The solution existence is a question of a delicate matter, and was solved only in a few particular cases. The existence of solutions is equivalent to a set of integrability conditions. The integrability conditions for the case of the Heisenberg group can be found in [1], p. 53 (see Theorem 2.9.8): Theorem 1. Let X1 = ∂x− 2y∂t, X2= ∂y+ 2x∂t be the Heisenberg vector fields on R3. The system X
1f = a, X2f = b has a solution f ∈ F (R3) if and only if
X12b = (X1X2+ [X1, X2])a X22a = (X2X1+ [X2, X1])b.
The paper [2] uses complexes of differential operators and symmetry re-ductions from the Heisenberg and Engel distributions to provide integrability conditions for the Grushin and Martinet distributions.
A direct method to prove the integrability conditions for Grushin dis-tributions of step 2 and 3 as well as for Heisenberg-type disdis-tributions on 3-dimensional manifolds is presented in the paper [3].
From the analysis performed in the aforementioned papers we know that in the case of distributions where the step exhibits jumps the exactness conditions are non-classical, in the sense that there are some extra functions that appear in the integrability relations. This occurs in the case of Grushin and Martinet distributions and this behavior is expected for all distributions with nonconstant step. On the other side, for constant step distributions (e.g. Heisenberg, Engel, etc.) we expect classical integrability conditions that can be expressed only in terms of vector fields {Xj}.
3. Equivalent Formulations
This section reviews a few operators used in sub-Riemannian geometry. We shall apply them to provide equivalent formulations for the solution existence problem that was presented in section 2.
Let f ∈ F (Rn). The following definition provides two equivalent ways to measure the change of the function f along a horizontal distribution. Definition 1. The horizontal gradient of f is the horizontal vector field ∇hf defined by
g(∇hf, X) = X(f ), ∀X ∈ Γ(H).
The horizontal differential of f is a horizontal 1-form (a form acting on horizontal vector fields) denoted by dhf and defined by
dhf (X) = X(f ), ∀X ∈ Γ(H).
We note that g(∇hf, X) = dhf (X), for all X ∈ Γ(H). The previous operators can be represented intrinsically as follows.
Proposition 1. Let {ωj} be dual 1-forms on Rn associated with the basic vector fields {Xj}, i.e. ωj(Xi) = δji. We have
(i) ∇hf =PjXj(f )Xj (ii) dhf =PjXj(f )ωj.
Proof. (i) Writing with respect to the basis {X1, · · · , Xm} yields ∇hf =X j g(∇hf, Xj)Xj = X j Xj(f )Xj.
(ii) It suffices to check the formula on a basis of Hp at a point p. Applying both sides to Xi, the left side becomes
dhf (Xi) = Xi(f ) while the right side yields the same formula
X j Xj(f )ωj(Xi) = X j Xj(f )δji = Xi(f ).
The next definition states a dual terminology.
Definition 2. A horizontal vector field U ∈ Γ(H) is called conservative, if there is a function f ∈ F (M ) such that U = ∇hf .
A horizontal 1-form ω is called exact if there is a function f ∈ F (M ) such that ω = dhf .
It is worth noting that there is a bijective correspondence between con-servative horizontal vector fields U and exact horizontal 1-forms ω; this correspondence is implied by the formula
g(U, X) = ω(X), ∀X ∈ Γ(H).
The system (2.1) can be written equivalently as ∇hf = U , with U = P
jajXj. Therefore, the fact that U is conservative is equivalent to the existence of solutions for the system (2.1).
On the other hand, the existence of solutions for the system (2.1) is equiv-alent to the exactness of a horizontal 1-form: given a horizontal 1-form η =P
jaj(x)ωj, find f ∈ F (Rn) such that dhf = η.
4. Main Result
The goal of this section is to obtain a result on the Heisenberg group that is an analog of the vectorial formula (1.6), and which will be given by Theorem 2. The Heisenberg distribution H is generated on R3 by the vector fields
X1 = ∂x− 2y∂z, X2 = ∂y+ 2x∂z.
Since X1, X2, [X1, X2] = 4∂z are linearly independent at every point, the distribution H is of constant step equal to 2 at each point. The dual coframe of X1, X2, X3= ∂z is given by
ω1= dx, ω2 = dy, ω3 = 2ydx − 2xdy + dz.
The differential of a function f can be written in terms of this coframe as df = X1(f )ω1+ X2(f )ω2+ X3(f )ω3
= df (X1)ω1+ df (X2)ω2+ df (X3)ω3.
Since the horizontal distribution is given by H = ker ω3, the constraint ω3 = 0 implies
dhf = X1(f )ω1+ X2(f )ω2.
The dual object to dhf is the horizontal gradient of f , which is given by ∇hf = X1(f )X1+ X2(f )X2.
A curve c(t) = x(t), y(t), z(t) is horizontal if the non-holonomic con-straint ω3( ˙c) = 0, i.e., 2y(t) ˙x(t) − 2x(t) ˙y(t) + ˙z(t) = 0 holds. In this case the velocity of the curve can be written as ˙c(t) = ˙x(t)X1+ ˙y(t)X2. The existence of smooth horizontal curves joining the origin with any given point (x, y, z) is proved, for instance, in [4] p. 10.
It is useful to note that if U = U1X1+ U2X2 and V = V1X1+ V2X2 are two horizontal vector fields and g denotes the sub-Riemannian metric, then g(U, V ) = U1V1+ U2V2.
Since [X1, X2] = 4∂z, the distribution is step 2 everywhere. The system X1f = a
X2f = b
can be written, after adding the additional condition [X1, X2]f = c, as ∂xf − 2y∂zf = a ∂yf + 2x∂zf = b (4.1) 4∂zf = c. Setting a1= a + 2y∂zf = a + yc/2 b1= b − 2x∂zf = b − xc/2 c1= c/4 = (X1b − X2a)/4 the system (4.1) becomes
∂xf = a1 ∂yf = b1 ∂zf = c1.
Consider the vector field V = (a1, b1, c1) on R3. We note that the equation ∇f = V has solutions if and only if curl V = 0.
Theorem 2. Let X1 = ∂x− 2y∂t, X2= ∂y+ 2x∂t be the Heisenberg vector fields on R3 and consider
(4.2) f (r) = Z 1 0 V (tr) · r dt. Then (X1f )(r) = a(r) + Z 1 0 tz 4 X12b − (X1X2+ [X1, X2] a)(tr) dt (X2f )(r) = b(r) − Z 1 0 tz 4 X22a − (X2X1+ [X2, X1])b (tr) dt. If the integrability conditions
X12b = (X1X2+ [X1, X2])a X22a = (X2X1+ [X2, X1])b hold, then X1f = a, X2f = b with f (x, y, z) = Z 1 0 g(Uγ(t), ˙γ(t)) dt,
Proof. We start by expressing the integrability conditions in terms of local coordinates. A straightforward computation provides
X12b − (X1X2+ [X1, X2])a = (−6∂z+ 4xy∂z2+ 2y∂y∂z− 2x∂x∂z− ∂x∂y)a + (4y2∂z2− 4y∂x∂z+ ∂x2)b,
(4.3)
X22a − (X2X1+ [X2, X1])b = (4x2∂z2+ 4x∂y∂z+ ∂y2)a
+ (6∂z+ 4xy∂z2+ 2y∂y∂z− 2x∂x∂z− ∂x∂y)b. (4.4)
Multiplying by the matrix A =
1 0 −2y
0 1 2x
on the left of the relation
(4.5) ∇f (r) = V (r) + Z 1 0 tr × curl V (tr) dt, we obtain A∇f (r) = AV (r) + Z 1 0 A tr × curl V (tr) dt ⇐⇒ X1f X2f (r) = a b (r) + Z 1 0
h(1, 0, −2y), tr × curl V (tr)i h(0, 1, 2x), tr × curl V (tr)i
dt. By the circular permutation invariance property of the mixed product of vectors, i.e., hu, v × wi = hw, u × vi, the previous formula becomes
(4.6) X1f X2f = a b + Z 1 0
hcurl V (tr), (1, 0, −2y) × tri hcurl V (tr), (0, 1, 2x) × tri
dt.
The explicit computation of the components of curl V = (curl V )1, (curl V )2, (curl V )3 is given in the following and will be used shortly
(curl V )1 = ∂yc − ∂zb = 1 4(−4x 2∂2 z− 4x∂y∂z− ∂y2)a +1 4(−6∂z− 4xy∂ 2 z − 2y∂y∂z+ 2x∂x∂z+ ∂x∂y)b; (curl V )2 = ∂za − ∂xc = 1 4(6∂z− 4xy∂ 2 z − 2y∂y∂z+ 2x∂x∂z+ ∂x∂y)a +1 4(−4y 2∂2 z + 4y∂x∂z− ∂x2)b; (curl V )3 = ∂xb − ∂ya = 1 2(6x∂z+ 2xy∂y∂z+ y∂ 2 y + 2x2∂x∂z+ x∂x∂y)a +1 2(6y∂z+ 2y 2∂ y∂z+ 2xy∂x∂z− y∂x∂y− x∂x2)b. Using the valuation of the mixed product as a determinant we have
hcurl V (r), (1, 0, −2y) × ri =
(curl V )1 (curl V )2 (curl V )3
1 0 −2y x y z
= 2y2(curl V )1− (z + 2xy)(curl V )2+ y(curl V )3 = z 4 n (−6∂z+ 4xy∂z2+ 2y∂y∂z− 2x∂x∂z− ∂x∂y)a + (4y2∂z2− 4y∂x∂z+ ∂x2)b o = z 4(X 2 1b − (X1X2+ [X1, X2])a)(r),
where we used (4.3). A similar computation applies to the second component
hcurl V (r), (0, 1, 2x) × ri =
(curl V )1 (curl V )2 (curl V )3
0 1 2x x y z
= (z − 2xy)(curl V )1+ 2x2(curl V )2− x(curl V )3 = −z 4 n (4x2∂z2+ 4x∂y∂z+ ∂y2)a + (6∂z+ 4xy∂z2+ 2y∂y∂z− 2x∂x∂z− ∂x∂y)b o = −z 4 X22a − (X2X1+ [X2, X1])b
by formula (4.4). Substituting back into (4.6) yields (X1f )(r) = a(r) + Z 1 0 tz 4 X12b − (X1X2+ [X1, X2] a)(tr) dt (X2f )(r) = b(r) − Z 1 0 tz 4 X22a − (X2X1+ [X2, X1])b (tr) dt. Let (4.7) f (r) = Z 1 0 V (tr) · r dt
be the work done by a force given by the vector field V along the curve {tr}, 0 ≤ t ≤ 1, and assume curl V = 0. Then ∇f = V on R3.
We review next a few vectorial calculus notions. Let c : [0, 1] → Rn be a smooth curve and Y be a smooth vector field. The fundamental theorem of calculus states that if Y = ∇f , then the integral depends only on the end point values of f Z c Y · dc = Z 1 0 hY, c0(t)i dt = f c(1) − f c(0).
Assume curl V = 0, so the system becomes ∇f = V on R3. Choosing c(t) = tr = (tx, ty, tz) we have the solution given by
f (r) = Z 1 0 V (tr) · r dt = Z 1 0 hV, c0(t)i dt = f c(1) − f c(0). The result holds for any other curve with the same end points.
If γ(t) = x(t), y(t), z(t) is a horizontal curve, then ˙γ = ˙xX1+ ˙yX2 and hence ˙z = 2x ˙y − 2 ˙xy. Let U = aX1 + bX2 be the horizontal vector field associated with our system. Then
Z 1 0 hV, γ0(t)i dt = Z 1 0 a1x(t) + b˙ 1y(t) + c˙ 1z(t)˙ dt = Z 1 0 (a +1 2yc) ˙x(t) + (b − 1 2cx) ˙y(t) + 1 4c ˙z(t) dt = Z 1 0 (a ˙x + b ˙y) dt + c 4 Z 1 0 (2y ˙x − 2x ˙y + ˙z | {z } =0 ) dt = Z 1 0 g(Uγ(t), ˙γ(t)) dt,
where g(·, ·) stands for the sub-Riemannian metric. Hence, if the integrabil-ity conditions hold, i.e., if
X12b = (X1X2+ [X1, X2])a X22a = (X2X1+ [X2, X1])b,
then the solution is given by the work done by the horizontal vector field U = aX + bY along any horizontal curve γ(t) = x(t), y(t), z(t) with c(0) = (0, 0, 0), c(1) = (x, y, z), and this is given by
f (x, y, z) = Z 1
0
g(Uγ(t), ˙γ(t)) dt.
This represents the work done by the force U along any horizontal curve
joining the origin and (x, y, z).
5. Acknowledgment
This research project was initiated when the second author visited the National Center for Theoretical Sciences, Hsinchu, Taiwan during January-April 2013 and the final version of the paper was completed while the second and the third author visited NCTS during July 2014. They would like to express their profound gratitude to the Director of NCTS, Professor Winnie Li for her invitation and for the warm hospitality extended to them during their stay in Taiwan.
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[10] S.L. Yap, The Poincar´e Lemma and an Elementary Construction of Vector Potentials, Amer. Math. Monthly, vol. 116, no. 3, (2009), 261-267.
E-mail address: ocalin@emich.edu
Department of Mathematics, Eastern Michigan University, Ypsi-lanti, Michigan 48197, USA
E-mail address: chang@georgetown.edu
Department of Mathematics, Department of Computer Science, George-town University, Washington D.C. 20057, USA and Department of Mathematics, Fu Jen Catholic University, Taipei 242, Taiwan, ROC
E-mail address: email: majhu@ust.hk
Department of Mathematics, Hong Kong University of Sciences and Technology, Kowloon, Hong Kong
