Geodesics in the Heisenberg group

We provide a new and elementary proof for the structure of geodesics in the Heisenberg group $\mathbb{H}^n$. The proof is based on a new isoperimetric inequality for closed curves in $\mathbb{R}^{2n}$. We also prove that the Carnot-Carath\'eodory metric is real analytic away from the center of the group.

We call HH n = span{X 1 , . . . , X n , Y 1 , . . . , Y n } the horizontal distribution on H n , and denote by H p H n the horizontal space at p. An absolutely continuous curve Γ : [0, S] → R 2n+1 is said to be horizontal ifΓ(s) ∈ H Γ(s) H n for almost every s ∈ [0, S]. It is easy to see that the horizontal distribution is the kernel of the standard contact form α = dt + 2 n j=1 (x j dy j − y j dx j ).
If the curve γ : [0, S] → R 2n is closed and Γ = (γ, t) is a horizontal lift of γ, then it follows from Green's Theorem that the total change in height t(S) − t(0) equals −4 times the sum of the signed areas enclosed by the projections γ j of the curve to each x j y j -plane. Since the curves in the x j y j -planes may have self-intersections, the signed area has to take multiplicity of the components of the complement of γ j into account. This multiplicity can be defined in terms of the winding number. We will not provide more details here as this interpretation of the change of the height will not play any role in our argument and it is mentioned here only to give an additional vantage point to the geometry of the problem.
We equip the horizontal distribution HH n with a left invariant Riemannian metric so that the vector fields X j and Y j are orthonormal at every point p ∈ H n . Note that this metric is only defined on HH n and not on T H n .
and hence c = 2 n j=1 (a j y j (p) − b j x j (p)). Clearly |v| H ≤ |v| E , where are the lengths of v with respect to the metric in H p H n and the Euclidean metric in R 2n+1 respectively. On compact sets in R 2n+1 the coefficient c is bounded by C|v| H , so for any Its length with respect to the metric in HH n equals (ẋ j (s) 2 +ẏ j (s) 2 ) ds.
The Carnot-Carathéodory metric d cc in H n is defined as the infimum of lengths ℓ H (Γ) of horizontal curves connecting given two points. It is well known that any two points can be connected by a horizontal curve (we will actually prove it) and hence d cc is a true metric. It follows from (1.2) that for any compact set K ⊂ R 2n+1 there is a constant C = C(K) such that If Γ : [a, b] → X is a (continuous) curve in a metric space (X, d), then the length of Γ is defined by where the supremum is taken over all n ∈ N and all partitions a = s 0 ≤ s 1 ≤ . . . ≤ s n = b.
In particular if Γ : [a, b] → H n is a curve in the Heisenberg group, then its length with respect to the Carnot-Carathéodory metric is It follows immediately from the definition that if Γ is a horizontal curve, then ℓ cc (Γ) ≤ ℓ H (Γ); hence every horizontal curve is rectifiable (i.e. of finite length), but it is not obvious, whether ℓ cc (Γ) = ℓ H (Γ). It is also not clear whether every rectifiable curve can be reparametrized as a horizontal curve. Actually all this is true and we have Proposition 1.1. In H n we have (1) Any horizontal curve Γ is rectifiable and ℓ cc (Γ) = ℓ H (Γ).
(3) Every rectifiable curve admits a 1-Lipschitz parametrization (and hence it is horizontal with respect to this parametrization).
Thus up to a reparametrization, the class of rectifiable curves coincides with the class of horizontal ones. Proposition 1.1 is well known, but it is difficult to find a good reference that would provide a straightforward proof. For the sake of completeness, we decided to provide a proof of this result in the Appendix. For a proof of (1) in more generality, see Theorem 1.3.5 in [15].
A geodesic from p to q is a curve of shortest length between the two points. It is already well known that any two points in the Heisenberg group can be connected by a geodesic. Also, the structure of every geodesic in a form of an explicit parameterization is known. The proofs in the case of H 1 can be found in [2,5,8,14] and the general case of H n is treated in [1,3,16].
Notice that the length ℓ H (Γ) of a horizontal curve Γ = (γ, t) given by (1.3) equals the usual Euclidean length ℓ E (γ) in R 2n of the projection γ.
If n = 1, the structure of geodesics can be obtained via the two dimensional isoperimetric inequality (see [2,5,14]). Consider a horizontal curve Γ = (γ, t) in H 1 connecting the origin to some point q = (0, 0, T ) with T = 0. The length of Γ equals the length of its projection γ to R 2 (which is a closed curve). Also, by (1.1), the change T in the height of Γ must equal −4 times the signed area enclosed by γ. Thus the projection of any horizontal curve connecting 0 to q must enclose the same area |T |/4, and finding a geodesic which connects 0 to q reduces to a problem of finding a shortest closed curve γ enclosing a fixed area. Thus the isoperimetric inequality implies that Γ will have smallest length when γ is a circle. Then the t component of Γ is determined by (1.1) and one obtains an explicit parametrization of the geodesics in H 1 connecting the origin to a point on the t-axis. Such geodesics pass through all points (x 0 , y 0 , t 0 ), t 0 = 0 in H 1 . If q = (x 0 , y 0 , 0), then it is easy to see that the segment 0q connecting the origin to q is a geodesic. This describes all geodesics connecting the origin to any other point in H 1 . Due to the left-invariance of the vector fields X and Y , parameterizations for geodesics between arbitrary points in H 1 may be found by left multiplication of the geodesics discussed above.
This elegant argument however, does not apply to H n when n > 1 and known proofs of the structure of geodesics in H n are based on the Pontryagin maximum principle, [1,3,16]. The purpose of this note is to provide a straightforward and elementary argument leading to an explicit parameterization of geodesics in H n (Theorem 2.1). Our argument is based on Hurwitz's proof [11], of the isoperimetric inequality in R 2 involving Fourier series. See [17] for a similar argument in a different but related setting. As an application of our method we also prove that the Carnot-Carathéodory metric is real analytic away from the center of the group, see Theorem 3.1. This improves a result of Monti [15,16] who proved that this distance is C ∞ smooth away from the center. We also find a formula for the Carnot-Carathéodoty distance (Corollary 3.2) that can be used to compute the terms of the power series expansion explicitly.
The paper is organized as follows. In Section 2 we will state and prove the result about the structure of the geodesics in H n , and in Section 3 we use this structure to show that the distance function in H n is analytic away from the center. In Section 4 we address the problem of comparing different geodesics (there are infinitely many of them) connecting the origin to a point (0, 0, T ) ∈ H n on the t-axis. Finally, in the Appendix we prove Proposition 1.1.
We hope that the paper will be of interest to those who are new to the subject of the Heisenberg group and for that reason we decided to make the paper self-contained.

The structure of geodesics
Any horizontal curve Γ is rectifiable and using arc-length parametrization we can assume that the speed |Γ| H of Γ : [0, ℓ H (Γ)] → H n equals 1. Then we can reparametrize it as a curve of constant speed defined on [0, 1], so we can assume that Γ : [0, 1] → H n satisfies On the other hand any rectifiable curve in H n can be reparametrized as a horizontal curve via the arc length parameterization (Proposition 1.1) and hence when looking for length minimizing curves (geodesics) it suffices to restrict to horizontal curves Γ : [0, 1] → H n satisfying (2.1).
Since the left translation in H n is an isometry, it suffices to investigate geodesics connecting the origin 0 ∈ H n to another point in H n . Indeed, if Γ is a geodesic connecting 0 to p −1 * q, then p * Γ is a geodesic connecting p to q.
If q belongs to the subspace R 2n × {0} ⊂ R 2n+1 = H n , then it is easy to check that the straight line Γ(s) = sq, s ∈ [0, 1] is a unique geodesic (up to a reparametrization) connecting 0 to q. Indeed, it is easy to check that Γ is horizontal and its length ℓ cc (Γ) = ℓ H (Γ) equals the Euclidean length |0q| of the segment 0q, because Γ is equal to its projection γ. For any other horizontal curveΓ = (γ,t) connecting 0 to q the projectionγ on R 2n would not be a segment (since horizontal lifts of curves are unique up to vertical shifts) and hence we would have ℓ cc (Γ) = ℓ H (Γ) = ℓ E (γ) > |0q| = ℓ cc (Γ) which proves thatΓ cannot be a geodesic.
In Theorem 2.1 we will describe the structure of geodesics in H n connecting the origin to a point (0, 0, T ) ∈ R 2n ×R = H n , T = 0, lying on the t-axis. Later we will see (Corollary 2.3) that these curves describe all geodesics in H n connecting 0 to q ∈ R 2n × {0}. The geodesics connecting 0 to q ∈ R 2n × {0} have been described above.
for j = 1, 2, . . . , n and Observe that if Γ(1) = (0, 0, +T ) the equations (2.2) give a constant-speed parametrization of negatively oriented circles in each of the x j y j -planes, centered at (A j , B j ), and of radius A 2 j + B 2 j . Each circle passes through the origin at s = 0. The signed area of such a circle equals −π(A 2 j + B 2 j ). Thus the change in height t(1) − t(0) which is −4 times the sum of the signed areas of the projections of the curve on the x j y j -planes equals Clearly this must be the case, because Γ connects the origin to (0, 0, T ). Any collection of circles in the x j y j -planes passing through the origin and having radii r j ≥ 0 are projections of a geodesic connecting the origin to the point (0, 0, T ) where T = 4π n j=1 r 2 j . In particular we can find a geodesic for which only one projection is a nontrivial circle (all other radii are zero) and a geodesic for which all projections are non-trivial circles. That suggests that the geodesics connecting (0, 0, 0) to (0, 0, T ) may have many different shapes. This is, however, an incorrect intuition. As we will see in Section 4 all such geodesics are obtained from one through a rotation of R 2n+1 about the t axis. This rotation is also an isometric mapping of H n . The above reasoning applies also to the case when Γ(1) = (0, 0, −T ) with the only difference that now the circles are positively oriented.
Proof of Theorem 2.1. Suppose first that Γ : [0, 1] → H n is any horizontal curve of constant speed connecting the origin to the point (0, 0, +T ), In particular the functions x j and y j are L-Lipschitz continuous and x j (0) = y j (0) = x j (1) = y j (1) = 0. Hence the functions x j , y j extend to 1-periodic Lipschitz functions on R and so we can use Fourier series to investigate them. We will follow notation used in [6]. By Parseval's identity, Thus by noting thatẋ j (s) =ẋ j (s) andẏ j (s) =ẏ j (s), we may apply Parseval's theorem to this pair of inner products and find The last equality follows from the identity |a + bi| 2 = |a| 2 − 2 Im(āb) + |b| 2 which holds for all a, b ∈ C. Since every term in this last sum is non-negative, it follows that L 2 4π 2 − T 4π ≥ 0. Thus, we have L 2 ≥ πT . Now Γ is a geodesic if and only if L 2 = πT which happens if and only if each of the two sums in (2.5) equals zero. Since k 2 − |k| > 0 for |k| ≥ 2, the first of the two sums vanishes if and only ifx j (k) =ŷ j (k) = 0 for every |k| ≥ 2 and j = 1, 2, . . . , n. Hence nontrivial terms in the last sum correspond to k = ±1 and the last sum vanishes if and only ifx j (±1) = −i sgn(±1)ŷ j (±1). That is, for every j = 1, . . . , n, x j (s) =x j (−1)e −2πis +x j (0) +x j (1)e 2πis y j (s) =ŷ j (−1)e −2πis +ŷ j (0) +ŷ j (1)e 2πis .
If we write A j = −(x j (−1) +x j (1)) and B j = −(ŷ j (−1) +ŷ j (1)), then we have the desired parameterization which is the (0, 0, +T ) case of (2.2). Note that it follows directly from the definition of Fourier coefficients that the numbers A j , B j are real. Now the formula for the t component of Γ follows from (1.1); the integral is easy to compute due to numerous cancellations. Remark 2.2. Notice that the inequality L 2 ≥ πT acts as a sort of isoperimetric inequality for curves in R 2n . Since the change in height T of the geodesic Γ is related to the sum A of the signed areas of the projected curves via the relationship T = −4A as seen in (1.1), our inequality becomes L 2 ≥ 4π|A|. In the case n = 1, this is the classical isoperimetric inequality known since antiquity.
To have a better understanding of the structure of geodesics it is convenient to use complex notation following the identification of R 2n with C n given by R 2n ∋ (x, y) = (x 1 , . . . , x n , y 1 , . . . , y n ) ↔ (x 1 + iy 1 , . . . , x n + iy n ) = x + iy ∈ C n .
With this notation the geodesics from Theorem 2.1 connecting the origin to (0, 0, ±T ), T > 0 can be represented as The question now is how to describe geodesics connecting the origin to a point q which is neither on the t-axis nor in R 2n × {0}. It turns out that geodesics described in Theorem 2.1 cover the entire space H n \ (R 2n × {0}) and we have Corollary 2.3. For any q ∈ H n which is neither in the t-axis nor in the subspace R 2n ×{0} there is a unique geodesic connecting the origin to q. This geodesic is a part of a geodesic connecting the origin to a point on the t-axis.
Proof. Let q = (c 1 , . . . , c n , d 1 , . . . , d n , h) be such that h = 0 and c j , d j are not all zero. We can write q = (c + id, h) ∈ C n × R. First we will construct a geodesic Γ given by (2.9) so that Γ(s 0 ) = q for some s 0 ∈ (0, 1). Clearly such a geodesic is a part of a geodesic connecting the origin to a point on the t-axis. Then we will prove that this is a unique geodesic (up to a reparametrization) connecting the origin to q. Assume that h > 0 (the case h < 0 is similar). We will find a geodesic passing through q that connects (0, 0, 0) to (0, 0, T ), for some T > 0. (If h < 0 we find Γ that connects (0, 0, 0) to (0, 0, −T ).) It suffices to show that there is a point A + iB ∈ C n such that the system of equations has a solution s 0 ∈ (0, 1). We have A + iB = (c + id)/(1 − e −2πis ) and hence This equation has a unique solution s 0 ∈ (0, 1) because the function on the left hand side is an increasing diffeomorphism of (0, 1) onto (0, ∞). We proved that, among geodesics connecting (0, 0, 0) to points (0, 0, T ), there is a unique geodesic Γ q passing through q. Suppose now thatΓ is any geodesic connecting (0, 0, 0) to q. GluingΓ with Γ q [s 0 ,1] we obtain a geodesic connecting (0, 0, 0) to (0, 0, T ) and hence (perhaps after a reparametrization) it must coincide with Γ q . This proves uniqueness.
We will now use the proof of Corollary 2.3 to find a formula for the Carnot-Carathéodory distance between (0, 0) and q = (z, h), z = 0, h > 0. We will need this formula in the next section. Let (2.12) H(s) = 2πs − sin(2πs) 1 − cos(2πs) be the diffeomorphism of (0, 1) onto (0, ∞) described in (2.11). Let be the geodesic from the proof of Corollary 2.3 that passes through q at s 0 ∈ (0, 1). We proved that s 0 is a solution to (2.11) and hence s 0 is a function of q given by .

Analyticity of the Carnot-Carathéodory metric
The center of the Heisenberg group H n is Z = {(z, h) ∈ H n | z = 0}. It is well known that the distance function in H n is C ∞ smooth away from the center [15,16], but, via the above arguments, we will now see that this distance function is in fact real analytic.
Theorem 3.1. The Carnot-Carathéodory distance d cc : R 2n+1 ×R 2n+1 → R is real analytic on the set Proof. In the proof we will make a frequent use of a well known fact that a composition of real analytic functions is real analytic, [13, Proposition 2.2.8]. It suffices to prove that the function d 0 (p) = d cc (0, p) is real analytic on H n \ Z. Indeed, w(p, q) = q −1 * p is real analytic as it is a polynomial. Also, d cc (p, q) = (d 0 • w)(p, q), so real analyticity of d 0 on H n \Z will imply that d cc is real analytic on w −1 (H n \Z) = {(p, q) ∈ H n ×H n | q −1 * p / ∈ Z}.
Define H : (−1, 1) → R as  Here, we divided by a common factor of (2πs) 2 in the two power series on the right hand side. That is, the denominator equals (1 − cos(2πs))(2πs) −2 which does not vanish on (−1, 1). This implies that H is real analytic on (−1, 1). Indeed, considering s as a complex variable, we see that H(s) is holomorphic (and hence analytic) in an open set containing (−1, 1) as a ratio of two holomorphic functions with non-vanishing denominator.
The function z → |z| −2 is analytic on R 2n \ {0} (as a composition of a polynomial z → |z| 2 and an analytic function 1/x), so the function (z, h) → h|z| −2 is analytic in H n \ Z. Hence also s 0 (q) = H −1 (h|z| −2 ) is analytic on H n \ Z.
In the case h = 0, Γ is a straight line in R 2n from the origin to q, and so d 0 (q) = |z|.
One can easily use this result to compute the terms of the power series expansion of (z, h) → d cc ((0, 0), (z, h)), z = 0.

Classification of non-unique geodesics
Any point (0, 0, ±T ), T > 0 on the t axis can be connected to the origin by infinitely many geodesics. The purpose of this section is to show that actually all such geodesics are obtained from one geodesic by a linear mapping which fixes the t-axis. This map is an isometry of H n and also an isometry of R 2n+1 . Proof. Consider geodesics Γ 1 = (γ 1 , t) and Γ 2 = (γ 2 , t) defined in the statement of the proposition. As in the discussion before Corollary 2.3, we consider γ 1 and γ 2 as functions into C n rather than into R 2n and write γ 1 (s) = 1 − e ∓2πis (A + iB), γ 2 (s) = 1 − e ∓2πis (C + iD) where 4π|A+iB| 2 = 4π|C +iD| 2 = T . We claim that there is a unitary matrix U ∈ U(n, C) such that U(A + iB) = C + iD. Indeed, for any 0 = z ∈ C n , use the Gram-Schmidt process to extend {z/|z|} to an orthonormal basis of C n and define W z to be the matrix whose columns are these basis vectors. Here, we consider orthogonality with respect to the standard Hermitian inner product u, v C = n j=1 u j v j . Then W z ∈ U(n, C) and W z e 1 = z/|z| where {e 1 , . . . , e n } is the standard basis of C n . Thus the desired operator is for every s ∈ [0, 1] and since V fixes the t-component of C n × R, we have V • Γ 1 = Γ 2 .
We now prove that V is an isometry on H n . Indeed, suppose p, q ∈ H n and Γ = (γ, t) : [0, 1] → H n is a geodesic connecting them. Then V • Γ = (U • γ, t). Since Γ is horizontal, it is easy to check thatṫ(s) = 2 Im γ(s),γ(s) C for almost every s ∈ [0, 1]. Unitary operators preserve the standard inner product on C n , and sȯ t(s) = 2 Im γ(s),γ(s) C = 2 Im (U • γ)(s), (U •γ)(s) C for almost every s ∈ [0, 1]. That is, V • Γ is horizontal. Also, Thus d cc (V p, V q) ≤ ℓ H (Γ) = d cc (p, q). Since U is invertible and U −1 ∈ U(n, C), we may argue similarly to show that d cc (p, q) = d cc (V −1 V p, V −1 V q) ≤ d cc (V p, V q), and so V is an isometry on H n . Clearly unitary transformations of C n are also orientation preserving isometries of R 2n and hence V is a rotation of R 2n+1 about the t-axis.

Appendix
Proof of Proposition 1.1. (1) The components of any horizontal curve Γ are absolutely continuous, and so their derivatives are integrable. Thus the inequality ℓ cc (Γ) ≤ ℓ H (Γ) < ∞ yields rectifiability of horizontal curves. It remains to prove that ℓ cc (Γ) ≥ ℓ H (Γ). Extend the Riemannian tensor defined on the horizontal distribution HH n to a Riemannian tensor g in R 2n+1 . For example we may require that the vector fields X j , Y j , T are orthonormal at every point of R 2n+1 . The Riemannain tensor g defines a metric d g in R 2n+1 in a standard way as the infimum of lengths of curves connecting two given points, where the length of an absolutely continuous curve α : [a, b] → R 2n+1 is defined as the integral This is the same approach that was used to define the Carnot-Carathéodory metric. For horizontal curves Γ, we have ℓ g (Γ) = ℓ H (Γ), and so it is obvious that d g (p, q) ≤ d cc (p, q) since we now take an infimum over a larger class of curves. It is a well known fact in Riemannian geometry that for an absolutely continuous curve α : where the supremum is taken over all n ∈ N and all partitions a = s 0 ≤ s 1 ≤ . . . ≤ s n = b as before. Hence if Γ : (2) If Γ : [a, b] → H n is Lipschitz, then by (1.4) it is also Lipschitz with respect to the Euclidean metric in R 2n+1 . Thus it is absolutely continuous and differentiable a.e. It remains to show thatΓ(s) ∈ H Γ(s) H n a.e. We will actually show that this is true whenever Γ is differentiable at s.
Suppose Γ is differentiable at a point s ∈ (a, b) and let i 0 ∈ N be such that s + 2 −i 0 ≤ b. Then d cc (Γ(s + 2 −i ), Γ(s)) ≤ L2 −i for some L > 0 and all i ≥ i 0 . By the definition of the Carnot-Carathéodory metric on H n , there is some horizontal curve η i : [0, 2 −i ] → H n which connects Γ(s) to Γ(s+2 −i ) whose length approximates the distance between them. That is, we can choose a horizontal curve η i = (x i , y i , t i ) so that η i (0) = Γ(s), η i (2 −i ) = Γ(s + 2 −i ) and ℓ H (η i ) < 2L2 −i . After a reparameterization, we may assume that η i has constant speed |η i | H < 2L on [0, 2 −i ]. Since η i is horizontal, we can writė η i (τ ) = n j=1ẋ i j (τ )X j (η i (τ )) +ẏ i j (τ )Y j (η i (τ )) for almost every τ ∈ [0, 2 −i ]. Now Γ(s + 2 −i ) − Γ(s) = η i (2 −i ) − η i (0) = 2 −i 0η i (τ ) dτ , and so i j (τ ) X j (η i (τ )) − X j (Γ(s)) dτ (5.1) The images of all the curves η i are contained in a compact subset of R 2n+1 . Hence (1. Here ℓ E (η i ) stands for the Euclidean length of η i . Observe also that the functionsẋ i j anḋ y i j are bounded almost everywhere by the speed |η i | H which is less than 2L. This and the continuity of the vector fields X j , Y j immediately imply that the sums (5.1) and (5.2) converge to zero as i → ∞. Also, by again using the uniform boundedness of the functionṡ x i j andẏ i j , we conclude that for some subsequence and all j = 1, 2, . . . , n the averages 2 i 2 −i 0ẋ i j and 2 i 2 −i 0ẏ i j converge to some constants a j and b j respectively. Denote such a subsequence by i k . Therefore, sinceΓ(s) exists we havė (a j X j (Γ(s)) + b j Y j (Γ(s))) ∈ H Γ(s) H n .
This proves (2). Finally (3) follows from a general fact that a rectifiable curve in any metric space admits an arc-length parametrization with respect to which the curve is 1-Lipschitz, see e.g. [4, Proposition 2.5.9], [10, Theorem 3.2].