An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group

We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is proved in terms of p-area which naturally arises from the variation of volume. The application makes a connection between CR geometry and integral geometry.


Introduction
In Euclidean spaces, the fundamental theorem of curves states that any unit-speed curve is completely determined by its curvature and torsion. More precisely, given two functions k(s) and τ (s) with k(s) > 0, there exists a unit-speed curve whose curvature and torsion are the functions k and τ , respectively, uniquely up to a Euclidean rigid motion. We present the analogous theorems of curves and surfaces in the 3-dimensional Heisenberg group H 1 . The main task in the paper also includes the understanding to the structure of the group of transformations in H 1 , which is similar to the group of rigid motions in Euclidean spaces. Moreover, we also develop the concept of the invariants for curves and surfaces in above sense. It should be emphasised that owning such invariants helps us understand the geometric structure in CR manifolds and develop the application to the field of Integral Geometry.
We give a brief review of the Heisenberg group. All the details can be found in [1]. The Heisenberg group H 1 is the space R 3 associated with the group multiplication (1.1) (x 1 , y 1 , z 1 ) • (x 2 , y 2 , z 2 ) = (x 1 + x 2 , y 1 + y 2 , z 1 + z 2 + y 1 x 2 − x 1 y 2 ), which is also a 3-dimensional Lie group. The standard left-invariant vector fields in H 1 The Heisenberg group H 1 can be regarded as a pseudo-hermitian manifold by considering H 1 associated with the standard pseudo-hermitian structure (J, Θ). Recall that a pseudo-hermitian transformation on H 1 is a diffeomorphism on H 1 preserving the pseudo-hermitian structure (J, Θ). For more information about pseudo-hermitian structure, we refer the readers to [3] [12][13] [18]. Denote P SH(1) be the group of pseudo-hermitian transformations on H 1 , and call the element in P SH(1) a symmetry. A symmetry in H 1 plays the same role as the rigid motion in R n and will be characterized in Subsection 3.1.
Let γ : I → H 1 be a parametrized curve. For each t ∈ I, the velocity γ ′ (t) has the natural decomposition where γ ′ ξ (t) and γ ′ T (t) are, respectively, the orthogonal projection of γ ′ (t) on ξ along T and the orthogonal projection of γ ′ (t) on T along ξ.
Definition 1.1. A horizontally regular curve is a parametrized curve γ(t) such that γ ′ ξ (t) = 0 for all t ∈ I. We say γ(t) is a horizontal curve if γ ′ (t) = γ ′ ξ (t) for all t ∈ I. From the approach of Contact Geometry, some authors call the horizontally regular cuvers the Legendrian curves, for examples, in [10,14,8]. Proposition 4.1 shows that a horizontally regular curve can always be reparametrized by the parameter s satisfying |γ ′ ξ (s)| = 1 for all s. Such a curve is called with horizontal unit-speed and the parameter s is called the horizontal arc-length for γ(s), which is unique up to a constant. Through the article the length of the vectors | · | and the inner product ·, · are defined with respect to the Levimetric, and the orthonormality of the vectorse 1 ande 2 on the contact plane ξ is always in the sense of the Levi-metric.
For a horizontally regular curve γ(s) parameterized by the horizontal arc-length s, we define the p-curvature k(s) and the contact normality τ (s) by where X(s) = γ ′ ξ (s) and Y (s) = JX(s). Notice that k(s) is analogous to the curvature of the curve in R n , while τ (s) measures how far the curve is from being horizontal. We also point out that k(s) and τ (s) are invariant under pseudo-hermitian transformations of horizontally regular curves.
The first theorem of the paper says that horizontally regular curves are completely prescribed by the functions k(s) and τ (s).
Theorem 1.2 (The fundamental theorem for curves in H 1 ). Given C 1 -functions k(s), τ (s), there exists a horizontally regular curve γ(s) with horizontal unit-speed having k(s) and τ (s) as its p-curvature and contact normality, respectively. In addition, any regular curve γ(s) with horizontal unit-speed satisfying the same p-curvature k(s) and contact normality τ (s) differs from γ(s) by a pseudo-hermitian transformation g ∈ P SH(1), namely, for all s.
Since a curve γ(s) is horizontal if and only if the contact normality τ (s) = 0, we immediately have the following corollary. Corollary 1.3. Given a C 1 -function k(s), there exists a horizontal curve γ(s) with horizontal unit-speed having k(s) as its p-curvature. In addition, any horizontal curve γ(s) with horizontal unit-speed satisfying the same p-curvature differs from γ(s) by a pseudo-hermitian transformation g ∈ P SH(1), namely, for all s.
If the horizontally regular curve γ is not parameterized by horizontal arc-length, we show, in Subsection 4.2, the explicit formulae for the pcurvature and the contact normality.
Theorem 1.4. Let γ(t) = x(t), y(t), z(t) ∈ H 1 be a horizontally regular curve, not necessarily with horizontal unit-speed. The p-curvature k(t) and the contact normality τ (t) of γ(s) are (1.8) Notice that in (1.8) the p-curvature k(t) depends only on x(t), y(t). We observe that k(t) is the signed curvature of the plane curve α(t) := π•γ(t) = x(t), y(t) , where π is the projection onto the xy-plane along the z-axis. It is the fact that the signed curvature of a given plane curve completely describes the curve's behavior, we have the corollary: Corollary 1.5. Suppose two horizontally regular curves in H 1 differ by a Heisenberg rigid motion, then their projections onto the xy-plane along the z-axis differ by a Euclidean rigid motion. In particular, two horizontal curves in H 1 differ by a Heisenberg rigid motion if and only if they are congruent in the Euclidean plane.
As an example, we calculate the p-curvature and contact normality for the geodesics, and obtain the characteristic description of the geodesics.
Theorem 1.6. In H 1 , the geodesics are the horizontally regular curves with constant p-curvature and zero contact normality.
In the second part of the paper, the fundamental theorem of surfaces in H 1 will be established. For an embedded regular surface Σ ⊂ H 1 , recall that a singular point p ∈ Σ is a point such that the tangent plane T p Σ coincides with the contact plane ξ p at p. Therefore outside the singular set (the non-singular part of Σ), the line bundle T Σ ∩ ξ forms a one-dimensional foliation, which is called the characteristic foliation.
where the norm is with respect to the Levi-metric. We call (u, v) normal coordinates of the surface F (U).
It is easy to see that normal coordinates always exist locally near a non-singular point p ∈ Σ. In addition, for a normal parameterized surface F , denote X = F u , Y = JX and T = ∂ ∂z . We define smooth functions a, b, c, l and m on U by and call a, b and c the coefficients of the first kind of F , and l, m the coefficients of the second kind. All coefficients satisfy the integrability conditions where the subscripts denote the partial derivatives.
The following theorem states that these coefficients are the complete differential invariants for the map F . and second kind of F , respectively. In addition, any F : U → H 1 normal parameterized surface with the same coefficients of first kind and second kind differ from F by a Heisenberg rigid motion, namely, In (5.24), we will show that the function l, up to a sign, is independent of the choice of normal coordinates, and hence it is a differential invariant of the surface F (U). Actually l is the p-mean curvature [4]. In particular, F (U) is a p-minimal surface when l = 0. Such a parametrization F : U → H 1 is called a normal p-minimal parameterized surface. In this case, the integrability condition (1.10) becomes (1.12) and the coefficients of first kind completely dominate those of second kind. We conclude all above as the following result. Theorem 1.9. Let U ⊂ R 2 be a simply connected open set. Suppose that a, b and c are smooth functions on U satisfying the integrability condition (1.11). Then there exists a normal p-minimal parameterized surface F : U → H 1 having a, b and c as the coefficients of first kind of F , which also determines the coefficient b of the second kind as in (1.12). In addition, any normal p-minimal parameterized surface F : U → H 1 with the same conditions differs from F by a Heisenberg rigid motion, namely, F (u, v) = g •F (u, v) in U for some g ∈ P SH(1).
In Section 5, we will obtain the other invariants on F (U): α := b c (up to a sign, called the p-variation), and the restricted adapted metric g Θ | Σ on the surface Σ. Actually α is the function such that the vector field αe 2 + T is tangent to the surface, where e 2 = Je 1 and e 1 is a unit vector field tangent to the characteristic foliation. Let e Σ be the unit vector field tangent to the surface then we observe that these invariants α, l, e Σ satisfy the integrability condition: (1 + α 2 ) 3 2 (e Σ l) = (1 + α 2 )(e 1 e 1 α) − α(e 1 α) 2 + 4α(1 + α 2 )(e 1 α) where K is the Gaussian curvature with respect to g Θ | Σ .
After studying the invariants in H 1 , we show the second main theorem which says that the three invariants: the Riemannian metric (induced by the adapted metric), the p-mean curvature, and the pvariation comprise a complete set of invariants for a surface without singular points.
Theorem 1.10 (The fundamental theorem for surfaces in H 1 ). Let (Σ, g) be a 2-dimensional Riemannian manifold with Guassian curvature K, and let α ′ , l ′ be two real-valued functions on Σ. Assume that K, α ′ and l ′ satisfy the integrability condition (1.13). Then for every nonsingular point p ∈ Σ, there exists an open neighborhood U containing p and an embedding f : U → H 1 such that where α, l are the induced p-variation and p-mean curvature on f (U). Moreover, f is unique up to a Heisenberg rigid motion.
The third part of the paper is an application of the motion equations and the structure equations we obtain for the proof of fundamental theorems. We will derive the Crofton formmula in H 1 which is a classic result of Integral Geometry [15] [16], relating the length of a fixed curve and the number of intersections for the curve and randomly oriented lines passing through it. In R 2 , given a fixed piecewise regular curve γ, the Crofton formula states that where dL is the kinematic density defined on the set of oriented lines in R 2 , and n(l ∩ γ) is the number of intersections of the line l with γ. We have the analogues formula in H 1 . Of particular interest is that the geometric quantity on one side is the p-area which naturally arises from the variation of the volume for domains in CR manifolds [4]. where dL := dp ∧ dθ ∧ dt is the kinematic density on L.
We give the outline of the paper. In section 2, we state two propositions about the existence and uniqueness of mappings from a smooth manifold into a Lie group G, which underlies our main theorems. In section 3, we not only express the representation of P SH(1) but discuss how the matrix Lie group P SH(1) can be interpreted as the set of moving frames on the homogeneous space H 1 = P SH(1)/SO(2); the moving frame formula in H 1 via the (left-invariant) Maurer-Cartan form will also be derived. In section 4, we compute the Darboux derivative of the lift of a horizontally regular curve and give the proof of the first main theorem; moreover, we calculate the p-curvature and the contact normality of horizontally regular curves and geodesics. In section 5, we compute the Darboux derivative of the lift of normal parameterized surfaces. By deriving the formula of change of coordinates, we achieve the complete set of differential invariants for a normal parameterized surface. In section 6, by calculating the Darboux derivative of the lift of f : Σ → H 1 , we show the fundamental theorem for surfaces Σ in H 1 . In section 7, one of the applications for the fundamental theorem of curves, the Crofton formula, will be shown, which connects CR geometry and Integral Geometry.
Acknowledgment. The first and second authors' research was supported in part by NCTS and in part by NSC 100-2628-M-008-001-MY4. He would like to thank Prof. Jih-Hsin Cheng for his teaching and talking with the author on this topic, Prof. Paul Yang for his encouragement and advising in the research for the last few years. The third author would like to express her thanks to Prof. Shu-Cheng Chang for his teaching, constant encouragement and supports.

Calculus on Lie groups
We recall two basic theorems from Lie groups, which play the essential roles in the proof of the main theorems. For the details we refer the readers to [2][7][10] [11] [17].
Given a connected smooth manifold M. Let G ⊂ GL(n, R) be the matrix Lie group with Lie algebra g and the (left-invariant) Maurer-Cartan form ω. We first introduce the theorem of uniqueness.
We call the Lie algebra-valued 1-form f * ω the Darboux derivative of the map f : M → G.
The second result is the theorem of existence.
Theorem 2.2. Suppose that φ is a g-valued 1-form on a simply connected manifold M. Then there exists a map f : Moreover, the resulting map f is unique up to a group action.
We mention that the proof of Theorem 2.2 relies on the Frobenius theorem.
3. The group of pseudo-hermitian transformations on H 1 3.1. The pseudo-hermitian transformations on H 1 . A pseudohermitian transformation on H 1 is a diffeomorphism Φ on H 1 preserving the CR structure J and the contact form Θ; it satisfies The trivial example of the pseudo-hermitian transformation is the left translation L p in H 1 ; the other example is defined by Φ R : where R ∈ SO(2) is a 2×2 special orthogonal matrix. Let P SH(1) be the group of pseudo-hermitian transformations on H 1 . We shall show that the group P SH(1) exactly consists of all the transformations of the forms Φ p,R := L p • Φ R , a transformation Φ R followed by a left translation L p . More precisely, we have Proof. It suffices to consider the pseudo-hermitian transformation Φ : then the composition L p −1 • Φ is a transformation fixing the origion. Therefore, we reduce the proof of Theorem 3.1 to the following Lemma: Let Φ be a pseudo-hermitian transformation on H 1 such that Φ(0) = 0. Then, for any p ∈ H 1 , the matrix representation of , for some real constant α 0 which is independent of p. Thus Φ * is a constant matrix.
To prove Lemma 3.2, we calculate the matrix representation of Φ * (p) with respect to the basis (e 1 ,e 2 , T ). For i = 1, 2, On the other hand, and, for all X ∈ ξ, By the uniqueness of the characteristic vector fields, we have Φ * T = T . From the above argument, we conclude that , for some real-valued function α on H 1 . Next, we rewrite the matrix representation of Φ * (p) from the basis e 1 ,e 2 , ∂ ∂z to the basis ∂ ∂x , ∂ ∂y , ∂ ∂z . Thus, and denote the subscripts as the partial derivatives for all Φ i 's. By (3.5) that Φ 1 z = Φ 2 z = 0, it follows that the functions Φ 1 and Φ 2 both depend only on x and y, and so is α. Moreover, use (3.5) again and the facts Φ 1 xy = Φ 1 yx and Φ 2 xy = Φ 2 yx , we have cos α − sin α sin α cos α α x α y = 0 0 , which implies that α x = α y = 0. Thus α is a constant on H 1 , say α = α 0 . From (3.5) and notice that Φ(0) = 0, we finally get . and the result follows. (1). The pseudo-hermitian transformation Φ p,R and the points (x, y, z) t in H 1 can be respectively represented as

Representation of P SH
Therefore, P SH(1) can be represented as a matrix group (3.10) Let psh(1) be the Lie algebra of P SH (1). It is easy to see that the element of psh(1) is of the form and the corresponding Maurer-Cartan form of P SH(1) is of the form where ω 2 1 and ω j , j = 1, 2, 3 are 1-forms on P SH(1). 3.3. The oriented frames on H 1 . The oriented frame (p; X, Y, T ) on H 1 consists of the point p ∈ H 1 and the orthonormal vector fields X ∈ ξ p , Y = JX with respect to the Levi metric. We can identify P SH(1) with the space of all oriented frames on H 1 as follows: (3.14) Actually, we have that X = ae 1 (p) + ce 2 (p) and Y = be 1 (p) + de 2 (p), hence M is the unique 4 × 4 matrix such that 3.4. Moving frame formula. Since P SH(1) is a matrix Lie group, the Maurer-Cartan form has to be ω = M −1 dM or dM = Mω (see [5]). Immediately one has that (3.16) (dp; dX, dY, dT ) = (p; X, Y, T ) Thus, we have reached the moving frame formula: where G is a Lie group, H is a closed Lie subgroup and G/H is a homogeneous space. In additional, the other liftF of f has to satisfỹ for some map g : M → H. By the moving frame formula (3.17), We also observe that all pull-back 1-forms by γ are the multiples of ds,  Insert γ * ω 3 into (3.17), we get As a consequence we obtain the Darboux derivative of γ   This completes the proof of the existence.
To prove the uniqueness, suppose that two horizontally regular curves γ 1 and γ 2 have the same p-curvature k(s) and contact normality τ (s). The identity (4.8) shows that they must have the same Darboux derivatives γ * 1 ω = γ * 2 ω. Therefore, by Theorem 2.1, there exists g ∈ P SH(1) such that γ 2 (s) = g • γ 1 (s), hence γ 2 (s) = g • γ 1 (s) for all s. This completes the proof of the uniqueness up to a group action.

4.2.
The derivation of the p-curvature and the contact normality. In the subsection, we will compute the p-curvature and the contact normality for horizontally regular curves (Theorem 1.4) and for the geodesics in H 1 (Theorem 1.6).
Proof of Theorem 1.4. Let γ(t) = (x(t), y(t), z(t)) be a horizontally regular curve. The horizontal arc-length s is defined by We first observe that there is the natural decomposition where we abuse the notations by ∂ ∂z = T . Letγ(s) be the reparametrization of γ(t) by the horizontal arc-length s. Since γ ′ (t) =γ ′ (s) ds dt , by comparing with the decomposition (4.12), one has For the p-curvature, by (4.13), note that X(s) = dt ds (x ′ (t)e 1 +y ′ (t)e 2 ), and Y (s) = JX(s) = dt ds (x ′ (t)e 2 − y ′ (t)e 1 ). A straight-forward computation shows so Again by (4.13), the contact normality has to be and the result follows.
Next we use (4.15) and (4.16) to compute the p-curvature and the contact normality for the geodesics in H 1 .
Proof of Theorem 1.6. Recall [1] that the Hamiltonian system on H 1 for the geodesics is So the Hamiltonian system (4.17) can be expressed bẏ Sinceξ 3 (t) = 0, we have ξ 3 (t) = c 3 for some constant c 3 . When c 3 = 0, one has x (t) = (c 1 t + d 1 , c 2 t + d 2 , (c 1 d 2 − c 2 d 1 ) t + d 3 ), and this implies k (t) = 0 and τ (t) = 0; when c 3 > 0, one has < 0 and τ (t) = 0; finally, when c 3 < 0, one has Hence k (t) = The calculations above show that a horizontal curve is congruent to a geodesic if it has positive constant p-curvature. Conversely, it is easy to prove that any geodesic acted by a symmetry is still a geodesic. Therefore we complete the proof of Theorem 1.6.
For the convenience, henceforward we simplify F (u, v) by F , and X(u, v) by X and so on. We first derive the Darboux derivative F * ω of F : By the moving frame formula (3.17), and apply on ∂ ∂u to get Compare the coefficients in (5.1)(5.2) and note that F u = X, we have Next we insert ∂ ∂v into (5.1) and compare the coefficients, one has  To derive F * ω 2 1 , we use (3.17) again and repeat the same process above. Since We obtain By (5.9), the coefficients a, b, c, l, m uniquely determine the Darboux derivative, and it completes the proof of the uniqueness.
Thus, φ satisfies the integrability condition dφ = −φ ∧ φ if and only lf the coefficients a, b, c, l and m satisfy the integrability condition (1.10). Therefore Theorem 2.2 implies there exists a map such that F * ω = φ. Finally, the moving frame formula (3.17) implies that F : U → H 1 is a map with a, b, c, l and m as the coefficients of first kind and second kind respectively.

5.2.
Invariants of surfaces. Let Σ ֒→ H 1 be a surface such that all points on Σ are non-singular. For each point p ∈ Σ, one can choose a normal parametrization F : (u, v) ∈ U → Σ around p such that where X is an unit vector field defining the characteristic foliation. The following lemma characterizes the normal coordinates.
Lemma 5.1. The normal coordinates is determined up to a transformation of the form for some smooth functions g(v), h(v) with ∂h ∂v = 0. Proof. Suppose that ( u, v) is any normal coordinates around p, i.e., (5.14) F u = X, where X = ±X. We have the formula for the change of the coordinates The first identity of (5.15) implies Since p is a non-singular point, we see that c = 0 around p, so As what did in the previous section (5.9), we can also derive the the Darboux derivative F * ω for the normal parametrization as. One obtains four 1-forms defined on Σ locally as follows: (5.20) where functions a, b, c, m and l are defined as (1.9). We will show that those four 1-forms are invariants under the change of coordinates.  ( u, v). We point out that all such the coefficients have the same expression as in (1.9) w.r.t. the new coordinates except for X = ±X and Y = J X = ±Y .
By lemma 5.1, there exists the functions g(v) and h(v) such that We compute the transformation laws of the coefficients of the fundamental forms then we have α = ± α. Actually, α is a function defined on the nonsingular part of Σ, independent of the chooses of the normal coordinates up to a sign, such that αe 2 + T ∈ T Σ, and hence an invariant of Σ on the non-singular part. Similarly, from (5.24), so is for l, which actually is the p-mean curvature.
Remark 5.4. We point out that the signs appearing for α and l are due to the different choices of the orientations. Indeed, if one chooses the normal coordinates with respect to a fixed orientation of the characteristic foliation, then we will have α = α and l = l.
Besides the invariants α and l, we now proceed the other invariant of Σ, which is globally defined on Σ, not just on the non-singular part. From Proposition 5.2, we have (5.27) I ⊗ I + II ⊗ II + III ⊗ III = I ⊗ I + II ⊗ II + III ⊗ III.
Therefore the differential form I ⊗ I + II ⊗ II + III ⊗ III again is independent of the choices of the normal coordinates, and hence also an invariant of Σ. Next we characterize the invariant.
Lemma 5.5. Let g Θ be the adapted metric on H 1 . Then we have on the non-singular part of Σ.
Proof. This lemma is a direct consequence of the moving frame formula (3.17).

5.3.
A complete set of invariants for surfaces in H 1 . In this section, we will obtain the last invariant IV = F * ω 2 1 which is completely determined by the invariants α, g Θ , l. We therefore have a complete set of invariants for the nonsingular part of the surfaces in H 1 .
Let Σ be an oriented surface and suppose f : Σ → H 1 be an embedding. For the convenience, we will not distinguish the surfaces Σ and f (Σ). For each non-singular point p ∈ Σ, we specify an orthonormal frame (p; e 1 , e 2 , T ), where e 1 is tangent to the characteristic foliation and e 2 = Je 1 . A Darboux frame is a moving frame which is smoothly defined on Σ except for the singular points, and hence there exists a lifting of f to P SH(1) defined by F . Now we would like to compute the Darboux derivative F * ω of F . In the following, instead of F * ω, we still use to express the Darboux derivative. It satisfies the integrability condition dω + ω ∧ ω = 0, that is, Let g Θ = h + Θ 2 be the adapted metric. From Section 5, we have ω 2 = αω 3 on the non-singular part of Σ, it is easy to see that This is an orthonormal coframe of g Θ | Σ and the corresponding dual frame isê 1 = e 1 , e 2 = e Σ = αe 2 + T √ 1 + α 2 . (5.32) Letω 2 1 be the Levi-Civita connection of g Θ | Σ with respect to the coframê ω 1 ,ω 2 . By the fundamental theorem in Riemannian geometry, we have the structure equation (5.33) The following Proposition shows that ω 2 1 is completely determined by the induced fundamental form g Θ | Σ and the functions α and l defined in (5.26).
7. The proof of Theorem 1.10 Proof. First we show the existence. Define an psh(1)-valued 1-form φ on the non-singular part of Σ by It is easy to check that φ satisfies dφ + φ ∧ φ = 0 if and only if the integrability condition (1.13) holds. Therefore, by Theorem 2.2, for each point p ∈ Σ there exists an open set U containing p and an embedding f : U → H 1 such that g = f * (g Θ ), α ′ = f * α and l ′ = f * l. For the uniqueness, by Proposition 5.6, the Darboux derivative is completely determined by the induced metric g Θ | Σ , the p-variation α and the p-mean curvature l. Therefore, by Theorem 2.1, the embedding into H 1 is unique up to a Heisenberg rigid motion.

Application: Crofton formula
Since the singular set in H 1 consists of isolated points and the integral over the set of isolated points has zero measure, we can always assume that there are no singular points in the context. Proof. We consider the projection π(l) of the line l ∈ L on the xyplane. Since π(l) can be uniquely determined by the pair (p, θ) where p ∈ R is the oriented distance from the origin to the line π(l) (see [6] or the Remark below) and θ ∈ [0, 2π) is the angle from the positive x-axis to the normal ( Fig. 1.1), the points (x, y) ∈ π(l) satisfy the equation x cos θ + y sin θ = p.
which carries (p, θ) to the line having the equation (8.2). It can be checked that X nonoriented is a two dimensional nonoriented smooth manifold equipping the two-folds covering spaces, more precisely one has if and only if either To our purpose, we consider the larger space equipping two orientations. (Fig. 1.2) Therefore, in H 1 , instead of using the nonoriented coordinates for the set  Next, we consider the intersections of lines and a fixed regular surface X : (u, v) ∈ Ω → (x(u, v), y(u, v), z(u, v)) ∈ Σ embedded in H 1 for some domain Ω ⊂ R 2 . To describe the position of the intersection in R 3 , one needs exact three variables. We have already known, by Proposition 8.2, a line can be represented by the triple (p, θ, t). Hence if we regard lines and surfaces as a whole system (the configuration space) and use five variables {(p, θ, t, u, v)} to describe the behavior of the intersections, two additional constraints are necessarily required to make the number of the freedoms be three. Those constraints can be obtained from the following Proposition.
Proposition 8.4. Let X(u, v) = (x(u, v), y(u, v), z(u, v)) ∈ Σ be the parameterized regular surface in H 1 . Then the configuration space D which describes the horizonal oriented lines intersecting Σ should be Proof. Suppose the line l(s) parameterized by (8.1) intersects the surface Σ at the point q. (Fig. 1.3) Fig. 1.3 At the point q, by Proposition 8.2, we have for some s ∈ R. By (8.7), (8.8), one has x(u, v) cos θ + y(u, v) sin θ = p, which is compatible with (8.2) and we obtain the first constraint (8.5). Finally, use (8.7),(8.8) again to solve for the parameter s, and substitute s into (8.9). It is easy to have the second constraint (8.6) Remark 8.5. By a simple calculation and (8.5), we observe that i.e. the horizontally unit-speed vector field U along the line have the same vector-value when being evaluated at the based point B and at the intersection q = X(u, v).
Actually, the coordinates (u, v) determine where the intersections should be located on the surface, and the angle θ decides how those lines penetrate through the surface. Thus, instead of using (p, θ, t) as the coordinates for the configuration space, we can also adopt the triple {(u, v, θ) ∈ Ω × S 1 } as the coordinates. Since the intersection q is not only on the line but on the surface, we can derive the change of the coordinates for those coordinates.
The next lemma characterize the 1-dimension foliation.
Lemma 8.7. Let E = αX u + βX v be the tangent vector field defined on the regular surface = X(u, v). Then the vector E is also on the contact bundle ξ (and hence in T H 1 ∩ ξ) if and only if pointwisely the coefficients α and β satisfy αt u + βt v + x(αy u + βy v ) − y(αx u + βx v ) = 0, (8.16) equivalently, Proof. First, we assume that E = αX u +βX v = α(x u , y u , z u )+β(x v , y v , z v ) = ce 1 + de 2 = (c, d, cy − dx) for some constants c and d. Compare each component of E to have αx u + βx v = c, αy u + βy v = d, αz u + βz v = cy − dx.
Substitute the last equation by the first two, we get the necessary condition αz u + βz v = (αx u + βx v )y − (αy u + βy v )x.
We have used the condition (8.16) in the last equality.
The second formula for the change of coordinates connects the coframes and the coordinates of the surface.
To prove the vector E ∈ T M ∩ ξ, it suffices to show that the coefficients α := (z u + xy u − yx u ) and β := −(z v + xy v − yx v ) satisfy the condition (8.17), and we complete the proof by the previous Lemma 8.7.
Thus, integrating the kinematic density dL over the set L, and use (8.15), (8.20), the Stock's theorem, and (8.21), which imply l∈L, l∩Σ =∅ n(l ∩ Σ)dL = 2 where Γ ± := ∂D ± \ Γ. We point out that for each line, there are two orientations passing through the surface, so we put double in front of the integral. Next, we show that du∧dv = 0 on Γ ± . Indeed, by using the coordinates {(u, v, θ)} for the configuration space D, any vector field defined on Γ + can be represented by A ∧ ∂ ∂θ ∈ ∂Σ × S 1 for some vector A defined on the tangent bundle T ∂Σ. The value du ∧ dv evaluated on Γ + has to be Therefore, (8.22) becomes l∈L, l∩Σ =∅ n(l ∩ Σ)dL = 2 2 we have used (8.21) and E is parallel to e 1 on Γ at the third equality.