Variational Calculus on Sub-Riemannian Manifolds

O. Călin and V. Mangione

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

We provide invariant formulas for the Euler-Lagrange equation associated to sub-Riemannian geodesics. They use the concept of curvature and horizontal connection introduced and studied in the paper.

Mathematics Subject Classification: 37J60, 53B21, 70H03 Key words: curvature, geodesics, connection

1 Introduction

The geodesic is a concept which comes from Riemannian geometry. It is the curve

with the minimum energy E = ∫_{0}^{1} \frac{1}{2}|ċ(s)|^{2} ds between two given points. At least

two kind of constraints can be considered to act on the curve: holonomic and nonholonomic. A holonomic constraint is when the energy is perturbed by a potential

U(c) and the energy becomes E = ∫_{0}^{1}(\frac{1}{2}|ċ|^{2} + U(c)) ds. The equation geodesic in this

case is ∇ċċ = −U′(c).

The other kind of constraints are the nonholonomic ones (see [1], [8], [9]). These are constraints on the velocity of the curve. The energy to be minimized is E =

= ∫_{0}^{1} (\frac{1}{2}|ċ|^{2}+ω(ċ)) ds. The paper deals with a presentation of the variational calculus

for the case when ω is a 1-form of type (1.1) such that (1.3) does not vanish. It is said that these kind of sub-Riemannian manifolds are of step 2. They are also called Heisenberg manifolds (see [2]). In general a sub-Riemannian manifold is said to be of step k if k − 1 iterations need for the brackets of X_{j} in order to span the whole tangent space.

In section 5 we shall deal with examples of sub-Riemannian manifolds of superior type.

The idea of the paper is to consider the solutions of the Euler-Lagrange system as geodesics in a certain connection with certain perturbation given by the curvature tensor defined in section 2. Section 3 shows that the classical Hamilton-Jacobi equation still holds if the gradient is modified into a horizontal gradient. The relationship between the symplectic and sub-Riemannian structures is pointed out in section 4. Section 5 provides a few examples of sub-Riemannian manifolds and their geodesic equations. Some of these equations were solved in [3,4,6].

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.1, 2003, pp. 21-32.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

Consider a nonintegrable 2-dimensional distribution x → H_{x} in R^{3} = R_{(x_{1},x_{2})}^{2}×R_{t} defined as H = ker ω, where ω is a 1-form on R^{3}. The distribution H is called the horizontal distribution. We shall assume the 1-form ω = ω^{1}dx_{1} + ω^{2}dx_{3} + ω^{3}dt has the coefficient ω^{3} ≠ 0 so that dividing by it we may assume

ω = −A_{1}(x)dx_{1} + A_{2}(x)dx_{2} + dt (1.1)

with A_{1} = −ω^{1}, and A_{2} = ω^{2}. One may verify that

ω(X_{1}) = ω(X_{1}) = 0

where X_{1} = ∂_{x_{1}} + A_{1}(x)∂_{t} , X_{2} = ∂_{x_{2}} − A_{2}(x)∂_{t} (1.2)

The vector fields X_{1}, X_{2} span the horizontal distribution H and they are called horizontal vector fields. Suppose the 2-form

Ω := dω = (\frac{∂A_{1}}{∂x_{1}} + \frac{∂A_{2}}{∂x_{1}} )dx_{1} ∧ dx_{2} (1.3)

does not vanish. Then

[X_{1}, X_{2}] = −(\frac{∂A_{1}}{∂x_{1}} + \frac{∂A_{2}}{∂x_{1}} )∂_{t} ∉ H (1.4)

and then H is not integrable, by Frobenius theorem. Consider the positive definite metric g : H × H → F in which the vector fields {X_{1}, X_{2}} are orthonormal. The metric g is called the sub-Riemannian metric defined by the vector fields X_{1} and X_{2}. A curve s → c(s) = (x_{1}(s), x_{2}(s), t(s)) is called horizontal curve if ċ(s) ∈ H_{c(s)}, for every s. As

ċ(s) = ẋ_{1}(s)∂_{x_{1}} + ẋ_{2}(s)∂_{x_{2}} + ṫ(s)∂_{t}
= ẋ_{1}(s)X_{1} + ẋ_{2}(s)X_{2} + ω(ċ(s))∂_{t}

then c(s) is a horizontal curve iff

ω(ċ) = ṫ − A_{1}(c)ẋ_{1} + A_{2}(c)ẋ_{2} = 0 (1.5)

The length of c with respect to the metric g is

l(c) = ∫_{0}^{1} \sqrt{g(ċ(s), ċ(s)} ds = ∫_{0}^{1} \sqrt{ẋ_{1}(s)^{2} + ẋ_{2}(s)^{2}} ds (1.6)

Given two points O and P there is at lest a horizontal curve connecting them (see [5]). The Carnot-Carathéodory distance is defined as

(1.7) d_{C}(O, P) = inf{l(c), c(0) = O, c(1) = P, c horizontal} Variational Calculus on Sub-Riemannian Manifolds 23 The horizontal curve with minimum length are called sub-Riemannian geodesics and can be described using the Hamiltonian formalism as in the following (see [7]). Consider the sub-elliptic operator

Δ_{X} = \frac{1}{2}(X_{1}^{2} + X_{1}^{2}) (1.8)

and define the Hamiltonian as the principal symbol of Δ_{X}

H(x, t, ξ, θ) = \frac{1}{2}(ξ_{1} + A_{1}(x)θ)^{2} + \frac{1}{2}(ξ_{2} − A_{2}(x)θ)^{2} (1.9)

The projections on the (x, t)-space of the solution of the Hamilton’s system ẋ = H_{ξ} , ṫ = H_{θ} (1.10) ξ̇ = −H_{x} , θ̇ = −H_{t} (1.11) with the boundary conditions x(0) = t(0) = 0 , x(1) = x, t(1) = t (1.12) are called sub-Riemannian geodesics between the origin and (x, t). From ṫ = H_{θ} we get ṫ = A_{1}ẋ_{1} − A_{2}ẋ_{2} (1.13) i.e. the sub-Riemannian geodesics are horizontal curves.

2 Connection and curvature

The horizontal connection

The horizontal connection is defined as D : H × H → H (2.14)

D(V, W) = D_{V} W = ∑_{k=1,2} V g(W, X_{k})X_{k} (2.15)

Proposition 2.1 D is a linear metric connection. Proof. One needs to verify the Leibnitz rule D_{V} (fW) = V (f)W + f D_{V} W (2.16) and the condition Ug(V, W) = g(D_{U}V, W) + g(V, D_{U}W) (2.17) For the first part,

D_{V} (fW) = ∑ V g(fW, X_{k}) X_{k} =
= ∑ V (f) g(W, X_{k})X_{k} + f ∑ V g(W, X_{k})X_{k} = V (f)W + fD_{V} W

24 O. Călin and V. Mangione To show the second part,

g(D_{U}V, W) + g(V, D_{U}W) =

= g( ∑ U g(V, X_{i})X_{i}, W ) + g(V, ∑ U g(W, X_{i})X_{i}) =
= g( ∑ U(V^{i})X_{i}, W ) + g(V, ∑ U(W^{i})X_{i}) =
= ∑ U(V^{i}) W^{i} + ∑ U(W^{i}) V^{i} = U( ∑ V^{i}W^{i}) = U g(V, W)

where V = ∑ V^{i}X_{i} and W = ∑ W^{i}X_{i}.

Let Z = Z^{1}X_{1} + Z^{2}X_{2} be a horizontal vector field. The horizontal divergence is defined as div_{H}Z = trace_{g}(V → D_{V} Z) = (2.18)

∑_{k} g(X_{k}, D_{X_{k}}Z) = ∑_{k} (X_{k}(Z^{j})X_{j})^{k} = ∑_{k} X_{k}(Z^{k}) = ∑_{k} X_{k} g(Z, X_{k}).

Define also the X-gradient of a function f as

∇_{X}f = ∑_{k} X_{k}(f) X_{k}. (2.19)

Then

\frac{1}{2}div_{H} ∇_{X} = Δ_{X}f (2.20)

The curvature tensor. Let K : H → H be given by

K(U) = ∑_{k} Ω(U, X_{k})X_{k}. (2.21)

K is F(R^{3})-linear and can be considered as a (1,1)-tensor of curvature.

The following result shows that K is skew-selfadjoint. Proposition 2.2 For every U, W ∈ H g(K(U), W ) + g(U, K(W)) = 0. (2.22) Proof. We show first that g(K(U), W ) = Ω(U, W), (2.23) and using the skew-symmetry of Ω we get (2.22).

Indeed,

g(K(U), W ) = g( ∑_{k} Ω(U, X_{k})X_{k}, W ) =

= ∑_{k} g(X_{k}, W)Ω(U, X_{k}) = Ω(U, W).

Variational Calculus on Sub-Riemannian Manifolds 25 Corollary 2.3 For any U ∈ H, g(K(U), U) = 0. (2.24) The last result suggests that in the case of a 2-dimensional distribution, the curvature K is proportional with a rotation of angle π/2. Define the rotation J : H → H as J (X_{1}) = X_{2} , J (X_{2}) = −X_{1} (2.25) Then

K(X_{1}) = Ω(X_{1}, X_{2})X_{2} = Ω(X_{1}, X_{2})J (X_{1})
K(X_{2}) = Ω(X_{2}, X_{1})X_{1} = Ω(X_{1}, X_{2})J (X_{2})

We arrived at the following formula for the curvature K(U) = Ω(X_{1}, X_{2})J (U), ∀U ∈ H (2.26) If the matrix Ω_{ij} is non-degenerate i.e. (\frac{∂A_{1}}{∂x_{1}} + \frac{∂A_{2}}{∂x_{1}} ) ≠ 0, then K(U) ≠ 0 for U ≠ 0.

If V is not a horizontal vector field then the curvature can still be defined using

K(V) = ∑_{k} Ω(V, X_{k})X_{k} (2.27)

This is because the right hand side depends only on the horizontal part of V. Indeed, consider the vector field

V = V^{1}∂_{x_{1}} + V^{2}∂_{x_{2}} + V^{3}∂_{t}

A computation shows

V = \underbrace{V^{1}X_{1} + V^{2}X_{2}}_{=V_{H}} +ω(V)∂_{t}

Then

Ω(V, X_{k}) = Ω(V_{H}, X_{k}) + ω(V) \underbrace{Ω(∂_{t}, X_{k})}_{=0}

Hence K(V) = K(V_{H}).

3 The Euler-Lagrange equation

The Legendre transform of the Hamiltonian (1.9) leads to the following Lagrangian

L(x, t, ẋ, ṫ) = \frac{1}{2}(ẋ_{1}^{2} + ẋ_{2}^{2}) + θ(ṫ − A_{1}(x)ẋ_{1} + A_{2}(x)ẋ_{2}), (3.28)

where θ is constant because 26 O. Călin and V. Mangione

θ̇ = −\frac{∂H}{∂t} = −\frac{dH}{dt} = 0. (3.29)

We deal now with a minimization problem with constraints given by

L(c, ċ) = \frac{1}{2}g(ċ, ċ) + θ ω(ċ) (3.30)

A computation shows the Euler-Lagrange system of equations

\frac{d}{ds} \frac{∂L}{∂ċ} = \frac{∂L}{∂c} , c = (x_{1}, x_{2}, t) (3.31)

becomes

ẍ_{1} = θ(\frac{∂A_{1}}{∂x_{2}} + \frac{∂A_{2}}{∂x_{1}} )ẋ_{2} (3.32)

ẍ_{2} = −θ(\frac{∂A_{1}}{∂x_{2}} + \frac{∂A_{2}}{∂x_{1}} )ẋ_{1} (3.33)

If the velocity of the geodesic is given by ċ(s) = ẋ_{1}(s)X_{1} + ẋ_{2}(s)X_{2}, the system (3.32) − (3.33) can be written as

ẍ_{1}X_{1} + ẍ_{2}X_{2} = θ(\frac{∂A_{1}}{∂x_{2}} + \frac{∂A_{2}}{∂x_{1}} )(ẋ_{2}X_{1} − ẋ_{1}X_{2}) (3.34)

The right hand side has the meaning of curvature. Indeed, using (2.25) and (2.26) the right hand side of (3.34) yields −θ Ω(X_{1}, X_{2}) J (ċ) = −θK(ċ). (3.35) For the left hand side of (3.34) consider the acceleration defined by the horizontal connection along ċ(s)

Dċċ = ∑_{k} ċ g(ċ, X_{k})X_{k} = ċ(ẋ_{1})X_{1} + ċ(ẋ_{2})X_{2} = ẍ_{1}X_{1} + ẍ_{2}X_{2}.

Hence the Euler-Lagrange system of equations can be written globally as Dċċ = −θK(ċ) (3.36) In sub-Riemannian geometry the acceleration of the geodesics is equal to the curvature. This keeps the geodesics into the horizontal distribution. Like in Riemannian geometry, we have Corollary 3.1 The length of velocity ċ in the sub-Riemannian metric g is constant. Proof. Since D is a metric connection,

ċ g(ċ, ċ) = 2g(Dċċ, ċ) = −2θg(K(ċ), ċ) = 0,

by Corollary 2.3.

Variational Calculus on Sub-Riemannian Manifolds 27

The Hamilton-Jacobi equation. Lemma 3.2 Let c(s) = (x_{1}(s), x_{2}(s), t(s)) be a horizontal curve and a smooth function f ∈ F(R^{3}). Then

\frac{df}{ds} = \frac{∂f}{∂s} + g(ċ, ∇_{X}f). (3.37)

Proof.

\frac{df}{ds} = \frac{∂f}{∂s} + \frac{∂f}{∂x_{1}} ẋ_{1} + \frac{∂f}{∂x_{2}} ẋ_{2} + \frac{∂f}{∂t} ṫ =
= \frac{∂f}{∂s} + (X_{1}f − A_{1}(x)\frac{∂f}{∂t} )ẋ_{1} + (X_{2}f + A_{2}(x)\frac{∂f}{∂t} )ẋ_{2} + \frac{∂f}{∂t} ṫ =
= \frac{∂f}{∂s} + (X_{1}f) ẋ_{1} + (X_{2}f) ẋ_{2} + \frac{∂f}{∂t} ω(ċ) = \frac{∂f}{∂s} + g(ċ, ∇_{X}f).

In the following we need to find the minimum of

I = ∫_{0}^{τ} \frac{1}{2}(ẋ_{1}(s)^{2} + ẋ_{2}(s)^{2}) ds = ∫_{0}^{τ} \frac{1}{2}|ċ(s)|_{g}^{2} ds

over the horizontal curves c(s) with fixed ends.

Let S(x, t) ∈ F be the solution for the Hamilton-Jacobi equation

\frac{∂S}{∂τ} + \frac{1}{2}|∇_{X}S|^{2} = 0, S(O) = 0. (3.38)

Consider the integral

J = ∫_{0}^{1} \frac{1}{2}|ċ(s)|_{g}^{2} ds − dS (3.39)

Using Lemma 3.2

\begin{aligned}J & = & ∫_{0}^{τ} (\frac{1}{2}|ċ(s)|_{g}^{2} − \frac{∂S}{∂s} − g(∇_{X}S, ċ)) ds = \\ & = & ∫_{0}^{τ} (\frac{1}{2}|ċ − ∇_{X}S|_{g}^{2} − (\frac{∂S}{∂s} + \frac{1}{2}|∇_{X}S|^{2})) ds = ∫_{0}^{τ} \frac{1}{2}|ċ − ∇_{X}S|_{g}^{2} ds \end{aligned} (3.40)

The integrals I and J reach the minimum for the same horizontal curve and this occurs for a curve with the velocity ċ = ∇_{X}S (3.41) Theorem 3.3 A horizontal curve c(s) is energy-minimizing iff (3.41) holds. Using (2.20) we get Corollary 3.4 The horizontal divergence of the geodesic flow is div_{H} ċ = 2Δ_{X}S (3.42)

The Hamiltonian. The Hamiltonian H : T^{∗}M → R is defined as

H(x, p) = \frac{1}{2} ∑_{k} p(X_{k})^{2}

If p = df,

H(x, df) = \frac{1}{2} ∑ df(X_{k})^{2} = \frac{1}{2} ∑ X_{k}(f)^{2} = \frac{1}{2}|∇_{X}f|^{2}.

For f = S,

H(x, dS) = \frac{1}{2}|∇_{X}S|^{2} = \frac{1}{2}|ċ|^{2} = \frac{1}{2}.

We also have

H(x, ω) = \frac{1}{2} ∑ ω(X_{i})^{2} = 0.

The eiconal equation. Consider the energy associated to a function f ∈ F(R^{3}) defined as

H(∇f) = H(x, df) = \frac{1}{2}|∇_{X}f|^{2} = \frac{1}{2}((X_{1}f)^{2} + (X_{2}f)^{2}) (3.43)

The front wave is given by the level curves of the energy and it is described by the eiconal equation

(3.44) H(∇f) = k, positive constant

with the initial condition

f(O) = 0 (3.45)

If k = 0, then f is the constant function equal to zero. Indeed, suppose that f is not constant. There is a point p such that (gradf)_{p} ≠ 0. Then Σ_{c} = f^{−1}(c) will be a surface through p, where c = f(p). As X_{i}(f) = 0, then X_{i} are tangent to Σ_{c} on a neighborhood of p and hence Σ_{c} becomes integral surface for the horizontal distribution H around p, which is nonintegrable, contradiction.

If k ≠ 0, consider the geodesics starting at origin c(0) = O, parametrized such that |ċ(s)|_{g}^{2} = 2k. If S is the action along c(s), by (3.41) we have

H(∇S) = \frac{1}{2}|∇_{X}S|_{g}^{2} = \frac{1}{2}|ċ|_{g}^{2} = k.

Jacobi vector fields and curvature. Let c(s) be a subRiemannian geodesic which starts at origin and let P be the first conjugate point with 0 along c(s). Denote by V (s) a Jacobi vector field along c(s) and by S(s) the action between 0 and c(s).

Proposition 3.5

∫_{0}^{1} K(V (s))(S(s)) ds = 0, (3.46)

where P = c(1) and K is the curvature.

Proof. Let c_{ϵ} = F_{ϵ}(c) be a smooth variation of c, such that for every ϵ, c_{ϵ} is a sub-Riemannian geodesic. As c_{ϵ} is a horizontal curve, then

0 = ∫_{0}^{1} ω(ċ_{ϵ}(s)) ds = ∫_{c_{ϵ}} ω = ∫_{F_{ϵ}(c)} ω = ∫_{c} F_{ϵ}^{∗}ω

Then

\frac{d}{dϵ} ∫_{c} F_{ϵ}^{∗}ω = 0

or,

∫_{c} L_{V} ω = 0,

where V is the Jacobi vector field associat to the variation (c_{ϵ})_{ϵ}. As V is zero at the end points of c,

∫_{c} d(i_{V} ω) = ∫_{∂c} i_{V} ω = ω(V)(0) − ω(V)(1) = 0.

Cartan decomposition yields

L_{V} ω = d(i_{V} ω) + i_{V} (dω),

and then

∫_{c} i_{V} Ω = 0,

which can also be written as

∫_{0}^{1} Ω(V (s), ċ(s)) ds = 0.

Using ċ = ∑ ċ^{j}X_{j} and ċ^{j}(s) = X_{j}(S), then

Ω(V, ċ) = Ω(V, ċ^{j}X_{j}) = ċ^{j}Ω(V, X_{j}) = Ω(V, X_{j}) X_{j}(S) = K(V)(S).

Hence

∫_{0}^{1} K(V)(S) ds = 0.

4 Constant curvature flow

In this section we ask the problem of a vector field such that |K(V)|^{2} = 1. As a nondegenerate, closed 2-form, Ω can be regarded as a symplectic form. One may associate the horizontal Hamiltonian vector field X_{f} to a function f as

Ω(X_{f}, W) = W (f), ∀W ∈ H (4.47)

We shall show that X_{f} has constant curvature for a certain f.

K(X_{f}) = ∑ Ω(X_{f}, X_{k})X_{k} = ∑ X_{k}(f)X_{k} = ∇_{X}f

and choosing f = S we have

|K(X_{S})|_{g}^{2} = |ċ|_{g}^{2} = 1 (4.48)

In the following we find the relation between the Hamiltonian field X_{S} and the geodesic flow ċ. Applying (2.26),

K(K(U)) = Ω(X_{1}, X_{2})K(J (U)) = Ω(X_{1}, X_{2})^{2}J^{2}(U) = −Ω(X_{1}, X_{2})^{2}U

or K^{2} = −Ω(X_{1}, X_{2})^{2}Id (4.49)

Using (4.49)

X_{S} = −Ω(X_{1}, X_{2})^{−2}K^{2}(X_{S}) = −Ω(X_{1}, X_{2})^{−2}K(ċ) = −Ω(X_{1}, X_{2})^{−1}J (ċ)

or ċ = Ω(X_{1}, X_{2})J (X_{S}), (4.50)

which provides the velocity of the geodesic in terms of the Hamiltonian vector field X_{S}.

5 A few examples of sub-Riemannian manifolds

5.1 The Heisenberg group IH_{1}.

The Heisenberg group constitutes the paradigm of the theory. The 3-dimensional Heisenberg group can be realized as H_{1} = R^{3} × R = {(x, t)} endowed with the group law (x, t) ∗ (x′, y′) = (x + x′, t + t′ + 2x_{2}x′_{1} − 2x_{1}x′_{2}). (5.51)

The vector fields which generate the nonintegrable distribution H are

X_{1} = ∂_{x_{1}} + 2x_{2}∂_{t} , X_{2} = ∂_{x_{2}} − 2x_{1}∂_{t}, T = ∂_{t}. (5.52)

They are left invariant with respect to the group law and generate the Lie algebra of H_{1}. The 1-form is ω = dt − 2x_{2}ẋ_{1} + 2x_{1}ẋ_{2} (5.53)

and the curvature 2-form is Ω = 4dx_{1} ∧ dx_{2} (5.54)

with Ω(X_{1}, X_{2}) = 4 (5.55)

and the curvature given by (2.26) becomes

K(U) = 4J (U), ∀U ∈ H (5.56)

The Euler-Lagrange equation is

ẍ = 4θJ (ẋ). (5.57)

Variational Calculus on Sub-Riemannian Manifolds 31

5.2 The (2n+1)-dimensional Heisenberg group IH_{n}.

The 2n-vector fields are defined on R^{2n+1} as

X_{k} = ∂_{x_{k}} + B_{k}(x)∂_{t} , k = 1, 2, … 2n (5.58) where

B_{j}(x) = ∑_{k=1}^{2n} 2a_{jk}x_{k} (5.59)

or B = 2Ax where A is a skew-symmetric non-singular matrix. The 1-form of connection in this case is ω = dt − 2Ax dx (5.60) Then the 2-form becomes

Ω = dω = 2 ∑_{p,j=1}^{2n} a_{pj}dx_{p} ∧ dx_{j} = −2⟨A dx, dx⟩ (5.61)

A computation shows that the curvature along a horizontal vector field U is

K(U) = − ∑_{k,p=1}^{2n} 4a_{pk}U^{k}X_{p} (5.62)

The Euler-Lagrange equation system of equations is given by

ẍ = −4θK(ẋ) (5.63)

5.3 A step 4 case.

Consider the vector fields

X_{1} = ∂_{x_{1}} + 4x_{2}|x|^{2}∂_{t}, X_{2} = ∂_{x_{2}} − 4x_{1}|x|^{2}∂_{t}, (5.64) which define the 1-form ω = dt − 4|x|^{2}(x_{2}dx_{1} − x_{1}dx_{2}). (5.65) Then Ω = 16|x|^{2}dx_{1} ∧ dx_{2}. (5.66) The curvature becomes K(U) = 16|x|^{2}J (U), ∀U ∈ H. (5.67) The Euler-Lagrange system is ẍ = 16θ|x|^{2}J (ẋ). (5.68)

5.4 A step 3 case.

The vector fields

X_{1} = ∂_{x_{1}} + \frac{x_{2}}{2} ∂_{t}, X_{2} = ∂_{x_{2}} (5.69)

define the Martinet distribution on R^{3}. Then

ω = dt − \frac{1}{2}x_{2}^{2}dx_{1} (5.70)

and Ω = x_{2} dx_{1} ∧ dx_{2} (5.71)

The curvature is K(U) = x_{2}J (U). (5.72)

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Eastern Michigan University Department of Mathematics Ypsilanti, MI, 48197, USA ocalin@emunix.emich.edu

Universitá degli Studi di Parma 43100, Parma, Via M. D’Azeglio, 85/A, Italia vittorio.mangione@unipr.it