LECTURE NOTES ON NON-HOLONOMIC GEOMETRY

JOSÉ RICARDO ARTEAGA, MIKHAIL MALAKHALTSEV

1. Distributions on manifolds

Let M be an n-dimensional smooth manifold. We denote by 𝔛(U) the Lie algebra of vector fields defined on an open U ⊂ M.

A k-dimensional distribution Δ on M is a smooth field of k-dimensional subspaces Δ(p) ⊂ T_{p}M, p ∈ M. The smoothness of this field means that, for any p ∈ M, there exists a neighbourhood U(p) and vector fields E_{1}, E_{2}, …, E_{k} ∈ 𝔛(U(p)) such that, for each q ∈ U(p), Δ(q) = span{E_{1}(q), E_{2}(q), …, E_{k}(q)}.

A distribution Δ on a manifold M is called totally integrable if for each p ∈ M there exists a submanifold Σ passing through p such that T_{q}Σ = Δ(q), for any q ∈ Σ. This submanifold is called an integral submanifold of Δ. A totally integrable distibution is called a foliation, and the integral submanifolds are called the leaves.

Locally, a distribution Δ can be given either by vector fields E_{1}, …, E_{k} spanning the subspaces Δ(p) ⊂ T_{p}M, or by differential forms ω^{1}, ω^{2}, …, ω^{n−k} such that, for any V ∈ T_{p}M, V ∈ Δ(p) if and only if ω^{α}(V) = 0, α = 1, …, n − k.

The following theorem gives necessary and sufficient conditions for a distribution be completely integrable.

Theorem 1 (Frobenius). Let Δ be a distribution on a manifold M locally given by vector fields E_{a}, a = 1, …, k, or by fields of 1-forms ω^{α}, α = 1, …, n − k. The following conditions are equivalent:

a) The distribution Δ is completely integrable;

b) [E_{a}, E_{b}] = Q_{ab}^{c}E_{c}, where Q_{ab}^{c} are functions;

c) dω^{γ} = Q_{β}^{γ} ∧ ω^{β}, where Q_{β}^{γ} are 1-forms.

Example 1.1. Consider a distribution Δ on ℝ^{3} given by ω = dz −xdx−ydy. Then dω = 0, and, by Theorem 1, Δ is completely integrable. In fact, this distribution consists of planes tangent to hyperbolic paraboloids z = \frac{1}{2}(x^{2} + y^{2}) + c.

Example 1.2. Let a distribution Δ on ℝ^{3} be given by ω = dy −zdx. Then the vector fields E_{1} = \frac{∂}{∂x} + z \frac{∂}{∂y}, E_{2} = \frac{∂}{∂z} span Δ. We have [E_{1}, E_{2}] = −\frac{∂}{∂y}, and this vector field does not lie in Δ because ω(\frac{∂}{∂y}) = −1 ≠ 0. Hence this distribution is not completely integrable. This means that one cannot find a surface tangent to Δ.

0This short course on non-holonomic geometry was delivered at workshop ”IV Taller Geometría Sub-Riemanniana y Operadores de Dirac en espacio-tiempo curvo” (September 24-28, 2007) in Universidad de Los Andes (Bogota, Colombia).

2. Non-holonomic surface in E^{3}

Let E_{3} be the three-dimensional Euclidean space, we denote by (·, ·) the scalar product in E_{3}. Any distribution Δ on E_{3} can be given by a unit vector field n⃗ normal to Δ. The integrability condition for Δ can be formulated in terms of n⃗.

Theorem 2. Δ is integrable if and only if (n⃗, rot n⃗) = 0.

2.1. Geometric sense of non-holonomity. The non-holonomity of a distribution in E^{3} can be visualized in the following manner. Take a small disk L in the distribution plane and draw the straight line l(A) through any point A of the disk boundary ∂L in the direction of n⃗(A). Thus we obtain a ruled surface. Now take a curve γ passing through A_{0} which is orthogonal to the straight lines l(A) (see Fig. 1). This curve again meets

Figure 1. Geometric sense of non-holonomity. I

l(A_{0}) at a point A_{1}, then \frac{|A_{0}A_{1}|}{σ(L)}, where σ(L) is the area of L, converges to (n⃗, rot n⃗)_{A_{0}} as L contracts to A_{0}. If Δ is holonomic (totally integrable), then γ lies on the integral surface passing through A_{0}, hence A_{1} = A_{0}, and |A_{0}A_{1}| = 0.

Another way to visualize the non-holonomity is to take two unit orthogonal vector fields a⃗, b⃗ spanning Δ. Let γ(t) be an integral curve of the field b⃗, and for each t we take the integral curve δ_{t}(s) of the field a⃗ passing through γ(t). Thus we obtain a surface Σ (see Fig. 2). Now at a point A(s) = δ_{t(s)}(s) we take the vector ν⃗(A(s)) normal to T_{A(s)}Σ and the vector n⃗(A(s)) normal to Δ(A(s)). Denote by φ(s) the angle between ν⃗(A(s)) and n⃗(A(s)). Then

(1)  \frac{dφ}{ds} = −(n⃗, rot n⃗).

If Δ is holonomic (totally integrable), then Σ is an integral manifold of Δ, and T_{p}Σ = Δ(p). Hence ν⃗(p) = n⃗(p), and φ = 0.

Figure 2. Geometric sense of non-holonomity. II

2.2. Curvatures of non-holonomic surface.

Figure 3. Moving frame

2.2.1. Curvatures of holonomic surface. Let a surface Σ in E_{3} be given by the parametric equation r⃗ = r⃗(u^{1}, u^{2}). Let us set r⃗_{i} = ∂r⃗_{i} = \frac{∂}{∂u^{i}} r⃗. The vectors r⃗_{1}, r⃗_{2} give a frame of the tangent plane T Σ, and the coordinates of the metric tensor (the first fundamental form) with respect to this frame are g_{ij} = (r⃗_{i}, r⃗_{j}). We denote by g^{ij} the tensor inverse to g_{ij}, this means that g^{is}g_{js} = δ_{j}^{i}. Now let n⃗ be the unit normal to T Σ. Then we obtain a moving frame {r⃗_{1}, r⃗_{2}, n⃗} (see Fig. 3) and the derivation equations:

∂_{i}r⃗_{j} = Γ_{ij}^{k}r⃗_{k} + h_{ij}n⃗ (2) ∂_{i}n⃗ = −h_{i}^{j}r⃗_{j} (3)

Here

(4)  Γ_{ij}^{k} = \frac{1}{2}g^{ks}(∂_{i}g_{js} + ∂_{j}g_{is} − ∂_{s}g_{ij})

4 JOSÉ RICARDO ARTEAGA, MIKHAIL MALAKHALTSEV

are the connection coefficients (we will discuss them later), which depend only on the metric tensor, h_{ij} is the second fundamental form, and h_{j}^{i} = g^{is}h_{js} is the shape operator.

The shape operator is a symmetric linear operator on T Σ, therefore it has real eigenvalues k_{1}, k_{2} which are called principal curvatures of Σ. H = \frac{1}{2}(k_{1} +k_{2}) is called the mean curvature of Σ and K = k_{1}k_{2} the total (Gaussian) curvature of Σ.

Geometrically, the principal curvatures can be described as follows. Take a point p ∈ Σ.

Figure 4. Normal curvature

For any direction tangent to Σ given by a unit vector τ⃗ at p, we can take the plane passing through n⃗ and τ⃗ (a normal plane), and this plane meets the surface Σ along a normal section γ(τ⃗). Let ρ⃗(s) be the parametric equation of γ(τ⃗) referred to the natural parameter s. Then, k_{n}(τ⃗) = (\frac{d^{2}}{ds^{2}} ρ⃗(s), n⃗) is called the normal curvature of Σ with respect to the direction τ⃗. As k_{n} is a continuous function on the circle, it attains its extremal values. One can easily prove that the principal curvatures k_{1} and k_{2} are just the extremal values of the normal curvature k_{n}(τ⃗). In addition, the normal curvature can be written in terms of the second fundamental form: k_{n}(τ⃗) = h(τ⃗, τ⃗). Also, using (3), we get

(5)  k_{n}(τ⃗) = −(τ^{i}∂_{i}n⃗, τ⃗) = −(∇τ⃗n⃗, τ⃗).

where ∇ stands for the directional derivative.

The total curvature can be defined in another way. The unit normal n⃗ determines the Gaussian map G : Σ → 𝕊^{2}, p ∈ Σ ↦ n⃗(p). Let U be a neighborhood of p ∈ Σ, and V = G(U) be its image on 𝕊^{2}. We denote by σ(U) the area of U on Σ, and by S(V) the area of V on 𝕊^{2}. Then the total curvature K at p equals lim \frac{S(V)}{σ(U)} as U contracts to p. From this property one can obtain another formula for the total curvature:

(6)  K = \frac{(∂_{1}n⃗, ∂_{2}n⃗, n⃗)}{\sqrt{g_{11}g_{22} − g_{12}^{2}}}

If we have a foliation in E^{3} whose leaves are given by an implicit equation Φ(x^{1}, x^{2}, x^{3}) = const, then the vector field n⃗ is defined in the entire E^{3}. In this case, from (6) it follows that

(7)  K = (P⃗ , n⃗),

where P⃗ is the curvature vector defined as follows:

(8)  P⃗ = ((\frac{∂}{∂x^{2}} n⃗, \frac{∂}{∂x^{3}} n⃗, n⃗), (\frac{∂}{∂x^{3}} n⃗, \frac{∂}{∂x^{1}} n⃗, n⃗), (\frac{∂}{∂x^{1}} n⃗, \frac{∂}{∂x^{2}} n⃗, n⃗)),

where (·, ·, ·) denotes the mixed product of three vectors. In particular, (7) implies the von Neumann formula:

K = \frac{| \begin{matrix}∂_{11}Φ & ∂_{12}Φ & ∂_{13}Φ & ∂_{1}Φ \\ ∂_{21}Φ & ∂_{22}Φ & ∂_{23}Φ & ∂_{2}Φ \\ ∂_{31}Φ & ∂_{32}Φ & ∂_{33}Φ & ∂_{3}Φ \\ ∂_{1}Φ & ∂_{2}Φ & ∂_{3}Φ & 0 \end{matrix} |}{(∂_{1}Φ)^{2} + (∂_{2}Φ)^{2} + (∂_{3}Φ)^{2})^{2}},

where the partial derivatives are taken with respect to x^{1}, x^{2}, x^{3}.

2.2.2. Curvatures of non-holonomic surface. First type. Let a two-dimensional distribution (a non-holonomic surface) Δ be given in E^{3}, and n⃗ be the unit normal vector field of Δ. The formula (5) gives us the normal curvature of a surface in terms of the normal vector n⃗ and a tangent vector τ⃗, so it can be generalized to the non-holonomic case. We say that the normal curvature of non-holonomic surface Δ in the direction of a unit vector τ ∈ Δ is

(9)  k_{n}(τ) = −(∇τ⃗n⃗, τ⃗).

Let A ∈ E^{3}, and e⃗_{1}, e⃗_{2} be an orthonormal frame of Δ(A). Then τ⃗(φ) = cos φe⃗_{1} + sin φe⃗_{2}, and one can easily see that k_{n}(τ⃗) defined by (9) has the form

(10)  k_{n}(τ⃗(φ)) = k_{1} cos φ + k_{2} sin φ,

and it is clear that k_{1} and k_{2} are extremal values of k_{n}(τ⃗(φ)). k_{1} and k_{2} are called the principal curvatures of Δ of first type. Then we can define the mean curvature of first type H_{I} = \frac{1}{2}(k_{1} + k_{2}) and the total curvature of first type K_{I} = k_{1}k_{2}.

2.2.3. Curvatures of non-holonomic surface. Second type. For a non-holonomic distribution Δ we can define the total curvature of second type using (7), where P⃗ is given by (8). We will denote it by K_{II}.

At the same time, this curvature can be obtained in the following way. Let us consider eigenvalues of the Jacobi matrix J(n⃗) = ||\frac{∂n^{i}}{∂x^{j}} ||, where n^{i} are the Cartesian coordinates of the vector-function n⃗. Since n⃗ maps the three-dimensional space into the two-dimensional sphere, the determinant of J(n⃗) is zero, hence one of the eigenvalues of J(n⃗) is zero. We will call the other two eigenvalues λ_{1}, λ_{2} the principal curvatures of second type (they can be complex conjugate) and one can prove that K_{II} = λ_{1}λ_{2}. Also, we can define the mean curvature of second type to be H_{II} = \frac{1}{2}(λ_{1} + λ_{2}).

Theorem 3. For a non-holonomic surface Δ we have

(11)  H_{I} = H_{II}  K_{I} − K_{II} = \frac{(n⃗, rot n⃗)^{2}}{4} .

Example 2.1. An example of distribution which has K_{II} = 0, H_{II} = 0, and is nonholonomic can be constructed as follows. We take the family of concentric spheres in E^{3} with center at the origin, and each sphere 𝕊^{2}(r) of the family is uniquely determined by its radius r > 0. For each r > 0, we take a straight line l(r) lying in the plane XOY such that the angle between l(r) and OX is α(r), where α(r) is a function (see Fig. 5). Now,

Figure 5. non-holonomic distribution with zero curvatures

for each 𝕊^{2}(r), we take the unit vector field n⃗ tangent to this sphere which is generated by rotations about the axis l(r) (certainly this field is not defined in the points where l(r) meets 𝕊^{2}(r)). Thus we obtain a unit vector field n⃗ which is defined everywhere in E^{3} except for points where l(r) meets 𝕊^{2}(r). If \frac{dα}{dr} ≠ 0, at points with z ≠ 0 we have (n⃗, rot n⃗) ≠ 0, so the distribution Δ orthogonal to n⃗ is non-holonomic. At the same time, the second type mean and total curvature of Δ vanish.

Example 2.2. Another way to obtain a distribution Δ with zero total curvature of second type is to consider a surface Σ with a vector field a⃗ along it (we assume that a⃗ is not tangent to Σ). Then, for each point p ∈ Σ we take a plane curve which is tangent to a⃗(p) at p. Thus, a neighborhood of Σ is foliated by these curves, and for the vector field n⃗ we take the unit vector field tangent to these curves. The distribution Δ orthogonal to n⃗ has K_{II} = 0.

2.2.4. Gauss-Bonnet formula. For a closed surface Σ in E^{3}, we have the Gauss-Bonnet formula

(12)  ∫_{Σ} KdS = 2πχ(Σ),

where K is the total curvature of Σ, and χ(Σ) is the Euler characteristic of Σ.

For a non-holonomic surface Δ, we have the following generalization of the Gauss-Bonnet formula.

Theorem 4. Let Q be a closed oriented surface which lies in the domain of Δ. Then

(13)  ∫_{Q}(K_{II}n⃗ + 2H_{II}k⃗ + ∇k⃗n⃗, ν⃗)dS = 4πθ,

where k⃗ is the curvature vector of integral curves of n⃗, ν⃗ is the unit normal to the surface Q, ∇k⃗ is the derivative in E^{3} with respect to k⃗, and θ is the degree of the map n⃗|_{Q} : Q → 𝕊^{2}.

Let us recall that the degree of a map f : Q → P between oriented surfaces is an integer which equals zero if f is homotopic to a constant map. Under assumptions of Theorem 4, Q is a boundary of a region V ⊂ E^{3}. If Δ is defined everywhere in V, and V is contractible, then n⃗ is homotopic to a constant vector field, hence θ = 0.

2.3. The shortest and the straightest curves on a non-holonomic surface. Let us recall that a geodesic on a surface Σ in the three-dimensional Euclidean space can be defined in two ways: a) a geodesic is a straightest curve; b) a geodesic is a locally shortest curve. Consider these definitions in more details.

Let us consider a curve on a surface Σ given by a parametric equation r⃗ = ρ⃗(s), where s is the natural parameter. The length of projection of the curvature vector \frac{d^{2}}{ds^{2}} ρ⃗ onto the plane tangent to Σ is called the geodesic curvature. If the geodesic curvature is zero, or equivalently the curvature vector is collinear to the normal vector of the surface, we say that this curve is a straightest curve on Σ because it has no “acceleration” along the surface.

At the same time, if a curve γ joining sufficiently close points p and q on the surface Σ has the minimal length among all the curves joining these points, then γ is called the shortest curve.

For a surface Σ ⊂ E^{3}, the definition a) is equivalent to definition b). However, for a non-holonomic surface Δ the curves defined by a) and b) are different. We will call a curve γ in E^{3} an admissible curve if its tangent vector \frac{d}{dt}γ lies in Δ(γ(t)).

2.3.1. Straightest lines on non-holonomic surfaces. Let a non-holonomic surface Δ be given. An admissible curve γ is called a straightest curve if its curvature vector is collinear to the normal vector n⃗ of Δ.

Theorem 5. Let r⃗ = ρ⃗(s) be parametric equation of an admissible curve γ. Then γ is a straightest line if and only if ρ⃗(s) satisfies the following second order differential equation:

(14)  \frac{d^{2}ρ⃗}{ds^{2}} = − (\frac{dρ⃗}{ds}, ∇_{\frac{dρ⃗}{ds}} n⃗) n⃗.

Thus, given a point p ∈ E^{3} and a vector V_{0} ∈ Δ(p) one can find a unique straightest curve γ which passes through p and whose tangent vector at p is V_{0}.

2.4. Shortest lines on non-holonomic surfaces. Let us consider two points p and q in E^{3}, and let P_{a}(p, q) be the set of all admissible curves joining p and q. We say that a curve γ ∈ P_{a}(p, q) such that γ(a) = p, γ(b) = q, is a shortest admissible curve if the length L(γ) = ∫_{a}^{b} |\frac{dγ}{dt} |dt is less than or equal to the length of any other curve in P_{a}(p, q).

Theorem 6. Let a curve γ ∈ P_{a}(p, q) be given by a parametric equation r⃗ = ρ⃗(s), s ∈ [a, b]. Then, if γ is a shortest curve, then ρ⃗(s) satisfies the third order ordinary differential equation

(15)  \frac{d^{2}ρ⃗}{ds^{2}} + \frac{d}{ds} \{\frac{(\frac{d^{2}ρ⃗}{ds^{2}}, \frac{dρ⃗}{ds}, n⃗)}{(n⃗, rot n⃗)}\} n⃗ + \frac{(\frac{d^{2}ρ⃗}{ds^{2}}, \frac{dρ⃗}{ds}, n⃗)}{(n⃗, rot n⃗)}[rot n⃗, \frac{dρ⃗}{ds}] = 0.

Therefore, given a point p ∈ E^{3}M and a vector V_{0} ∈ Δ(p) one can find an infinite number of shortest curves joining p and q which pass through p and whose tangent vector at p is V_{0}.

Figure 6. A unique straightest curve and a bundle of shortest curves

3. Properties of metric of sub-Riemannian manifold

3.1. Accessibility of points. Let M be an n-dimensional differential manifold, and Δ be an m-dimensional distribution on M. We will call a curve γ in M admissible if its

LECTURE NOTES ON NON-HOLONOMIC GEOMETRY 9

tangent vector lies in Δ. Let us denote by P_{a}(p, q) the set of all admissible curves joining p and q (this set can be empty!). We say that a point q is accessible from a point p if P_{a}(p, q) is not empty.

Theorem 7 (Sussman). The set of points accessible from a given point p is an immersed submanifold in M.

3.2. Completely non-integrable distributions. First let us define a distribution with singularities on an n-dimensional manifold M. A distribution with singularities Δ on M is a smooth field of subspaces Δ(p) ⊂ T_{p}M, p ∈ M. The smoothness of this field means that, for any p ∈ M, there exists a neighbourhood U(p) and vector fields E_{1}, E_{2}, …, E_{k} ∈ 𝔛(U(p)) such that, for each q ∈ U(p), Δ(q) = span{E_{1}(q), E_{2}(q), …, E_{k}(q)}. Note that we do not assume that the vectors E_{1}(q), E_{2}(q), …, E_{k}(q) are linearly independent (this makes difference from the definition of k-dimensional distribution, see Section 1). Certainly, each k-dimensional distribution is a distribution with singularities.

Now let Δ be a distribution with singularities on M. Let 𝔛_{Δ}(U) be the set of vector fields X on an open set U ⊂ M such that, for each q ∈ U, X(q) ∈ Δ(q). Now, for the distribution Δ we can construct another distribution Δ^{1} with singularities in the following way. For any point p ∈ M, we set

(16) Δ^{1}(p) = {X(p), [X, Y](p) | X, Y ∈ 𝔛_{Δ}(U_{p}), where U_{p} is a neighborhood of p}

From this definition it follows that Δ(p) ⊂ Δ^{1}(p), for any p ∈ M.

It is clear that, if Δ is completely integrable if and only if Δ^{1} = Δ.

For a distribution Δ we can sequentially construct the series Δ^{k} of distributions with singularities by setting Δ^{k+1} = (Δ^{k})^{1}. Since Δ^{k} ⊂ Δ^{k+1}, and Δ^{k} = Δ^{k+1} if and only if Δ^{k} is completely integrable, there exists N such that Δ^{N} = Δ^{N+1} = Δ^{N+2} = …, and Δ^{N} is completely integrable. Thus, if the subspaces Δ^{N}(p) do not coincide with T_{p}M, then we get a foliation (with singularities) on M. If, for each p ∈ M, we have Δ^{N}(p) = T_{p}M, then we say that the distribution Δ is completely non-integrable.

If a distribution Δ is completely non-integrable, then the following theorem implies that any point of M is accessible from any other point of M.

Theorem 8 (Caratheodory, Rashevskii, Chow). Let a distribution Δ on a connected manifold M be completely non-integrable. Then, for any p, q in M, P_{a}(p, q) ≠ ∅.

3.3. Sub-Riemannian metric of completely non-integrable distributions. A sub-Riemannian manifold is a completely non-integrable distribution Δ endowed by a metric g. This means that, for any p in M, on Δ(p) we have the scalar product g_{p} : Δ(p)×Δ(p) → ℝ such that for any X, Y ∈ 𝔛_{Δ}(M) the function p ↦ g_{p}(X(p), Y (p)) is smooth. In other words, we can say that we have a metric on Δ considered as a vector subbundle of the vector bundle T M.

If γ(s), s ∈ [a, b], is an admissible curve in M, then we can define the length of γ(s) as follows:

(17)  L(γ) = ∫_{a}^{b} g_{γ(s)}(\frac{dγ}{ds} , \frac{dγ}{ds})ds

From theorem 8 it follows that we can define a distance d on M:

(18)  d(p, q) = inf_{γ∈P_{a}(p,q)} L(γ).

This distance has some nice properties like the distance defined by a Riemannian metric. For example, we have the following theorem.

Theorem 9. The topology defined by the distance d coincides with the original topology on M.

A curve γ(s), s ∈ [a, b], is called a minimizing geodesic if L(γ) = d(γ(a), γ(b)).

Theorem 10 (Hopf-Rinow theorem for sub-Riemannian manifolds). a) Sufficiently near points can be joined by a minimizing geodesic;

b) If (M, d) is a complete metric space, then any two points can be joined by a minimizing geodesic.

4. References

The detailed exposition of results on geometry of non-holonomic surfaces in the threedimensional Euclidean space (stuff of section 2) can be found in the very interesting and profound book by Yu. Aminov:

Yu. Aminov: The geometry of vector fields. Amsterdam: Gordon and Breach Publishers. (2000).

For the introduction to the sub-Riemannian geometry (stuff of section 3), we can recommend the book:

R. Montgomery A Tour of Subriemannian geometries, Their Geodesics and Applications , Mathematical Surveys and Monographs, Vol. 91, AMS, 2002.