Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 71–74

THE INVERSION OF WHITEHEAD’S THEOREM

ON CONVEX REGIONS ^{1}

Lubomír Klapka

Abstract. In 1932 J. H. C. Whitehead [2] proved the existence of convex neighbourhoods with respect to the geodesic arcs corresponding to a given connection on a manifold. The present paper gives necessary and sufficient conditions for a set of paths to be the set of all geodesics of some connection. The main condition is the existence of convex neighbourhoods in a more general sense. Keywords. Linear connections, geodesics, convex regions. MS classification. 53B05, 53C05, 53C22.

Fundamental notions: We suppose that all manifolds and mappings used throughout this paper are smooth. We consider a closed one-dimensional interval ⟨0, 1⟩ ⊂ R and a n-dimensional manifold X. The derivatives with respect to the global coordinate on ⟨0, 1⟩ are denoted by dots. On X we use local coordinate systems

φ : V → φ(V) ⊂ R^{n}, V ⊂ X,  (1)

with coordinates Q, q or x, where for example Q = (Q^{1}, Q^{2}, …, Q^{n}) = (Q^{k}) ∈ φ(V) ⊂ R^{n}, k = 1, 2, …, n. On the tangent bundle T X with the natural projection π : T X → X we use local coordinate systems

T φ : T V → T φ(T V) = φ(V) × R^{n}  (2)

associated with (1) and denote the corresponding coordinates by (x, v) ∈ φ(V) × R^{n}. Everywhere we use the standard summation convention with the summation from 1 to n. The most important notions, which are considered in this work, are described in the following two paragraphs.

A path γ on X is a mapping γ : ⟨0, 1⟩ → X. Its coordinate expression is denoted by x(t) or q(t). Let P be a set of all polynomial mappings ⟨0, 1⟩ → ⟨0, 1⟩ whose degree is less than two, that is, the mappings of the form θ(τ) = (1 − τ)T + τt, where T, t ∈ ⟨0, 1⟩. We say that a path γ ∘ θ, where θ ∈ P, is a subpath of the path γ. We understand a linear connection Γ on X in a usual way given for example in [1]. Notice that a linear connection Γ on X exists if and only if there is, for each point c ∈ X, a local coordinate system (1), c ∈ V, and functions Γ_{lm}^{k}(x) on φ(V) satisfying, for every coordinate transformation φ̄ ∘ φ^{−1}|_{φ(V̄ ∩V)}, the relations

1This paper is in final form and no version of it will be submitted for publication elsewhere.

\frac{∂^{2}x̄^{k}}{∂x^{l}∂x^{m}} + Γ̄_{rs}^{k}(x̄)\frac{∂x̄^{r}}{∂x^{l}} \frac{∂x̄^{s}}{∂x^{m}} = \frac{∂x̄^{k}}{∂x^{r}} Γ_{lm}^{r}(x).  (3)

A geodesic of a linear connection Γ on X is a path each coordinate expression of which satisfies the differential equations

ẍ^{k}(t) + Γ_{lm}^{k}(x(t))ẋ^{l}(t)ẋ^{m}(t) = 0.  (4)

Let W (X) be a set of paths on X. We say that W (X) is closed on subpaths if and only if γ ∈ W (X), θ ∈ P implies γ ∘ θ ∈ W (X). We say that an open set U ⊂ X is convex with respect to W (X) if and only if

a) for each two points a, b ∈ U there exists a unique path γ_{ab} ∈ W (X) such that γ_{ab}(0) = a, γ_{ab}(1) = b, γ_{ab}(⟨0, 1⟩) ⊂ U,

b) the mapping ρ : U × U × ⟨0, 1⟩ ∋ (a, b, τ) → γ_{ab}(τ) = ρ(a, b, τ) ∈ U is smooth. We say that X is locally convex with respect to W (X) if and only if for each c ∈ X there exists an open set U such that c ∈ U ⊂ X, U is convex with respect to W (X). We say that W (X) is a maximal set of paths on X satisfying the conditions c_{1}, c_{2} if and only if for every set W̄ (X) ⊃ W (X) of paths on X satisfying the conditions c_{1}, c_{2} the relation W̄ (X) = W (X) holds.

Theorem: Let W (X) be a set of paths on a manifold X. The following two assertions are equivalent:

1) There is a linear connection Γ on X such that W (X) is a set of all its geodesics,

2) W (X) is a maximal set of paths on X satisfying the conditions:

a) X is locally convex with respect to W (X),

b) W (X) is closed on subpaths.

Lemma 1 (Whitehead): Let W (X) be a set of all geodesics of some linear connection Γ on a manifold X. Then X is locally convex with respect to W (X).

Proof^{2} of Lemma 1: Considering (3) we can define a vector field η on T X by the chart expression

(T T φ) ∘ η ∘ (T φ)^{−1} = v^{k} \frac{∂}{∂x^{k}} − Γ_{lm}^{k}(x)v^{l}v^{m} \frac{∂}{∂v^{k}}  (5)

in each of the local coordinate systems (2). Let exp(τη) be the local flow of this vector field, h ∈ ω, where ω is the zero section of π. From (5) we obtain exp(τη)(h) = h. So the local flow exp(τη) is defined on ω for each τ ∈ R. Then there is an open set M̄, ω ⊂ M̄ ⊂ T X, such that the flow exp(τη) : M̄ → T X is defined for each τ ∈ ⟨0, 1⟩.

Let us consider a point c ∈ X and a coordinate system (1) such that c ∈ V, φ(c) = 0. A matrix, whose components are A_{kl}(x) = g_{kl} − Γ_{kl}^{m}(x)g_{mr}x^{r}, where g_{kl} = 1 for k = l and g_{kl} = 0 for k ≠ l, is positive definite for x = 0. Hence there is an open set Ū, c ∈ Ū ⊂ V, such that the matrix, whose components are A_{kl}(φ(a)), is positive definite for each point a ∈ Ū.

2The proof of Lemma 1 is merely a simplification of the proof of Theorem 8.7 given in [1].

Let us consider a function χ : φ(V) ∋ x → g_{kl}x^{k}x^{l} ∈ ⟨0, ∞). From (5) the mapping σ̄ : M̄ ∋ m → (π(m), π ∘ exp(η)(m)) ∈ X × X is regular at a point h ∈ ω, π(h) = c, and π ∘ exp(τη)(h) = c for each τ ∈ ⟨0, 1⟩. Hence there are a real number δ > 0 and an open set M ⊂ X, h ∈ M ⊂ M̄, such that

a) φ(U), c ∈ U = {a ∈ Ū|χ ∘ φ(a) < δ} ⊂ X, is an open ball on φ(V),

b) σ : M ∋ m → (π(m), π ∘ exp(η)(m)) ∈ U × U is a diffeomorphism,

c) π ∘ exp(τη)(M) ⊂ Ū for each τ ∈ ⟨0, 1⟩. From (4) and (5) we have

d) the mapping γ_{ab} : ⟨0, 1⟩ → X, defined by

γ_{ab}(τ) = π ∘ exp(τη) ∘ σ^{−1}(a, b),  (6)

satisfies the conditions γ_{ab} ∈ W (X), γ_{ab}(0) = a, γ_{ab}(1) = b,

e) F̈(τ) = 2A_{kl}(x(τ))ẋ^{k}(τ)ẋ^{l}(τ) ≥ 0, where F = χ ∘ φ ∘ γ_{ab}, x(τ) = φ ∘ γ_{ab}(τ), which implies F (τ) ≤ (1 − τ)F (0) + τF (1) < (1 − τ)δ + τδ = δ and so γ_{ab}(⟨0, 1⟩) ⊂ U.

Because σ is a diffeomorphism, for each two points a, b ∈ U there exists the unique path (6) such that γ_{ab} ∈ W (X), γ_{ab}(0) = a, γ_{ab}(1) = b, γ_{ab}(⟨0, 1⟩) ⊂ U, and the mapping ρ : U × U × ⟨0, 1⟩ ∋ (a, b, τ) → γ_{ab}(τ) = ρ(a, b, τ) ∈ U is smooth. Consequently the open set U ⊂ X is convex with respect to W (X). In this way for each c ∈ X we obtain an open set U such that c ∈ U ⊂ X and U is convex with respect to W (X). This completes the proof of Lemma 1.

Lemma 2: Let W (X) be a set of all geodesics of some linear connection Γ on a manifold X. Then W (X) is closed on subpaths.

Proof of Lemma 2: Let γ ∈ W (X). Since the coordinate expression q(t) of γ satisfies the differential equations (4), the coordinate expression x(t) of γ ∘ θ, θ ∈ P, satisfies, by θ(t) = (1 − t)θ(0) + tθ(1), x(t) = q(θ(t)), ẋ(t) = q̇(θ(t))(θ(1) − θ(0)), ẍ(t) = q̈(θ(t))(θ(1) − θ(0))^{2}, the differential equations (4) too, and so γ ∘ θ ∈ W (X). Therefore, γ ∈ W (X), θ ∈ P implies γ ∘θ ∈ W (X) and the proof of Lemma 2 is finished.

Lemma 3: Let W (X) be a set of paths on X satisfying the conditions a), b) of the Theorem. Then there is a linear connection Γ on X such that W (X) is a set of its geodesics.

Proof of Lemma 3: Let an open set U ⊂ X be convex with respect to W (X), a, b ∈ U. Since W (X) is closed on subpaths, it holds that γ_{ab}(1 − τ) = γ_{ba}(τ), γ_{ab}(0) = γ_{ab}(0τ) = γ_{aa}(τ). From this and the definition of convexity, the smooth mapping ρ(a, b, τ) = γ_{ab}(τ) satisfies

ρ(a, b, 1 − τ) = ρ(b, a, τ),  (7)
ρ(a, a, τ) = a.  (8)

Let c ∈ X. From the condition a) of the Theorem there exist a coordinate system (1) and an open convex with respect to W (X) set U ⊂ X so that c ∈ V ⊂ U. Let D be a diagonal of V × V. From (8) there is an open set N, D ⊂ N ⊂ V × V, such that ρ(N × ⟨0, 1⟩) ⊂ V. We denote (ρ^{k}(Q, q, τ)) = φ ∘ ρ ∘ ψ^{−1}(Q, q, τ), where

ψ : N × ⟨0, 1⟩ ∋ (a, b, τ) → (φ(a), φ(b), τ) ∈ ψ(N × ⟨0, 1⟩) ⊂ R^{2n+1}.

From (7) and (8) by considering

(\frac{∂ρ^{k}(Q, q, \frac{1}{2})}{∂Q^{l}} + \frac{∂ρ^{k}(Q, q, \frac{1}{2})}{∂q^{l}})|_{Q=q=x}= \frac{∂(ρ^{k}(Q, q, \frac{1}{2})|_{Q=q=x})}{∂x^{l}}

it follows that

\frac{∂ρ^{k}(Q, q, \frac{1}{2})}{∂Q^{l}} |_{Q=q=x}= \frac{∂ρ^{k}(Q, q, \frac{1}{2})}{∂q^{l}} |_{Q=q=x}= \frac{1}{2}δ_{l}^{k},  (9)

where δ_{l}^{k} is the Kronecker symbol. So for each point c ∈ X there is a local coordinate system (1), c ∈ V, and functions

Γ_{lm}^{k}(x) = −4\frac{∂^{2}ρ^{k}(Q, q, \frac{1}{2})}{∂Q^{l}∂q^{m}} |_{Q=q=x}  (10)

on φ(V) satisfying, by (9) and (10), the relations (3). Therefore, the linear connection Γ on X, given by the relations (10), exists.

Let γ ∈ W (X), τ_{0} ∈ ⟨0, 1⟩. Then there are a closed interval I = ⟨τ_{1}, τ_{2}⟩, τ_{1} < τ_{2}, a coordinate system (1) and an open convex with respect to W (X) set U ⊂ X such that τ_{0} ∈ I ⊂ ⟨0, 1⟩, γ(I) ⊂ V ⊂ U ⊂ X. From the condition b) of the Theorem we have γ∘θ ∈ W (X), γ∘θ(⟨0, 1⟩) ⊂ U, γ∘θ(0) = γ(T), γ∘θ(1) = γ(t), where θ(τ) = (1−τ)T +τt, T, t ∈ I. From the condition a) of the Theorem we get γ ∘ θ(τ) = ρ(γ(T), γ(t), τ). The coordinate expression of this equation is

x^{k}((1 − τ)T + τt) = ρ^{k}(x(T), x(t), τ).  (11)

From (10) by differentiating (11) with respect to T, t and putting T = t, τ = \frac{1}{2} we obtain that x(t) satisfies the differential equations (4). Therefore, each path γ ∈ W (X) is a geodesic of the connection Γ on X. This completes the proof of Lemma 3.

Proof of the Theorem: Let 1) hold. Then from Lemma 1 and Lemma 2 W (X) satisfies the conditions a), b). Every set W̄ (X) ⊃ W (X) of paths on X satisfying the conditions a), b) satisfies, by Lemma 3, also the condition W̄ (X) ⊂ W (X), and so W̄ (X) = W (X). Therefore, W (X) is a maximal set of paths on X satisfying conditions a), b), and 2) holds.

Conversely, suppose 2). Then from Lemma 3 there is a linear connection Γ on X such that W (X) is a set of its geodesics. The set W̄ (X) ⊃ W (X) of all geodesics of the linear connection Γ satisfies, from Lemma 1 and Lemma 2, the conditions a), b). From the maximality property W (X) = W̄ (X) follows, and 1) holds. This completes the proof of the Theorem.

Remark: Let us suppose that the assertion 2) of the Theorem is satisfied. From the condition b) we have that the monoid of all mappings θ ∈ P acts on W (X) from right in the following way W (X) ∋ γ → γ ∘ θ ∈ W (X). The condition a) makes one possible to introduce a natural structure of a manifold on W (X), with respect to which the mentioned monoid is the Lie monoid.

Acknowledgment: The author is grateful to the referee for improvement of the formulation of the Abstract.

References:

[1] S. Kobayashi and K. Nomizu, Foundations of differential geometry (Interscience Publishers, New York, London, 1963), Volume I, 113–153.

[2] J. H. C. Whitehead, Convex regions in the geometry of paths, Quart. J. Math. 3 (1932), 33–42. Author’s address: Lubomír Klapka, Department of Mathematics, Silesian University at Opava,

Bezručovo nám. 13, 746 01 Opava 1, Czech Republic.

Received 24 September 1992 Revised 24 November 1992