Lobachevskii Journal of Mathematics Vol. 5, 1999, 13–28 \begin{matrix}c \\ ◯ \end{matrix}S.Z. Németh

S.Z. Németh

GEODESIC MONOTONE VECTOR FIELDS

(submitted by A.M.Elizarov)

Abstract. Having in mind the Minty-Browder monotonicity notion, we shall generalize it for vector fields on Riemannian manifolds, defining the geodesic monotone vector fields. The gradients of geodesic convex functions, important in optimization, linear and nonlinear programming on Riemannian manifolds, are geodesic monotone vector fields. The geodesic monotonicity will be related with the first variation of the length of a geodesic. The connection between the existence of closed geodesics and monotone vector fields will also be analyzed. We give a class of strictly monotone vector fields on a simply connected, complete Riemannian manifold with nonpositive sectional curvature, which generalize the notion of position vector fields. The notion of geodesic scalar derivative will be introduced for characterization of geodesic monotone vector fields on such manifolds. The constant sectional curvature case will be analized separately, since it has important peculiarities.

1 Introduction

The Minty-Browder monotonicity notion (see [MI], [ZE]) was an important achievement of nonlinear analysis, throwing a new light to many application oriented investigations. Although many results of convex analysis and optimization theory were generalized to Riemannian manifolds there are few attempts to generalize this notion. The most natural way we could thought of generalizing the monotone operators of Euclidean spaces to Riemannian manifolds is to introduce the notion of geodesic monotone vector fields (and connected to this the notion of strictly geodesic monotone vector fields, virtually geodesic monotone vector fields and trivially geodesic monotone vector fields). C. Udrişte proves [UD1] that a function f defined on an open and geodesic convex subset of a Riemannian manifold is geodesic convex, if and only if its gradient is geodesic monotone. This result is expressed, in terms of the differential of f, and is stated in [UD2] too. T. Rapcsák in [RA] reformulates the result of Udrişte, giving explicitly the inequality which expresses the geodesic monotonicity of the gradient of a geodesic convex function. However, as far as we know, nobody has been considering (in general) geodesic monotone vector fields yet. We shall prove that a vector field X on a Riemannian manifold is geodesic monotone (strictly geodesic monotone), if and only if the first variation of the length of every geodesic arc γ with infinitesimal variation the restriction of X to γ is nonnegative (positive). Analyzing the existence of geodesic monotone vector fields, we shall prove that there is no strictly geodesic monotone vector field on a Riemannian manifold which contains a closed geodesic. Similarly, if every geodesic of a Riemannian manifold is closed then there is no virtually monotone vector field on the manifold. As a consequence, since every compact Riemannian manifold contains a closed geodesic [KL], there are no strictly monotone vector fields on compact Riemannian manifolds. For the existence of strictly geodesic monotone vector fields, we consider simply connected, complete Riemannian manifolds with nonpositive sectional curvature, proving that the f-position vector fields of such manifolds are strictly geodesic monotone, where the f-position vector fields generalize the notion of position vector fields. We can generalize the notion of scalar derivative (see [NE1], [NE2]), introduced by us, for characterizing such vector fields. The results are generalizations of some results of [NE1] and [NE2]. The case of constant sectional curvature Riemannian manifolds presents some interesting peculiarities, therefore we shall consider it separately. For illustrating the ideas it is enough to consider just the case of S^{n}, where S^{n} is the n-dimensional sphere. The author expresses his gratitude to professor Tamás Rapcsák, professor János Szenthe, professor Gyula Soós, dr. Balázs Csikós and dr. Endre Szolcsányi for many helpful conversations.

2 Geodesic monotone vector fields on Riemmanian manifolds

Recall the following definition:

Definition 2.1 Let M be a Riemannian manifold.

(i) A subset K of M is called geodesic convex [RA] if for every two points of M there is a geodesic arc joining these points contained in G.

(ii) Let K be a geodesic convex subset of M. A function f : K → R is called geodesic convex (strictly geodesic convex) [RA], if f ∘ γ : [0, l] → R is convex (strictly convex) for every every unit speed geodesic arc γ : [0, l] → M contained in K.

If N is an arbitrary manifold, we shall denote by Sec(T N) the family of sections of the tangent bundle T N of N. We remark that Sec(T N) contains the uncontinuous vector fields too. Using this notation we have the following definition:

Definition 2.2 Let (M, ⟨ . , . ⟩) be a Riemannian manifold, K ⊂ M a geodesic convex open set and X ∈ Sec(T K) a vector field on K. X is called geodesic monotone if for every x, y ∈ K and every unit speed geodesic arc γ : [0, l] → M joining x and y (γ(0) = x, γ(l) = y) contained in K we have that

⟨X_{x}, γ′(0)⟩ ≤ ⟨X_{y}, γ′(l)⟩,

where γ′ denotes the tangent vector of γ with respect to the arclength. Let X be geodesic monotone. With the previous notations we have that: X is called strictly geodesic monotone if for every distinct x and y yields

⟨X_{x}, γ′(0)⟩ < ⟨X_{y}, γ′(l)⟩.

X is called virtually geodesic monotone if there are x and y such that

⟨X_{x}, γ′(0)⟩ < ⟨X_{y}, γ′(l)⟩.

X is called trivially geodesic monotone if for every x, y we have that

⟨X_{x}, γ′(0)⟩ = ⟨X_{y}, γ′(l)⟩.

It is easy to see that X is geodesic monotone (strictly geodesic monotone), if and only if for every unit speed geodesic γ the function

υ : t ↦ ⟨X_{γ(τ)}, γ′(τ)⟩

is monotone (strictly monotone). Similarly X is trivially monotone, if and only if υ is constant. The following example makes the connection between geodesic monotone vector fields and monotone operators of an Euclidean space, showing that with few modifications the formers are generalizations of the laters:

Example 2.3 Let E be an Euclidean space, G ⊂ E an open and convex set and h : G → E a monotone operator. Then the vector field X ∈ Sec(T G); x ↦ h(x)_{x}, where h(x)_{x} is the tangent vector in 0 of the curve t ↦ x + th(x), is geodesic monotone.

The following theorem is a modified version of Udrişte’s result [UD2], [UD1].

Theorem 2.4 Let M be a Riemannian manifold and K an open and geodesic convex subset of M. A function f : K → R is geodesic convex (strictly geodesic convex), if and only if its gradient grad f is geodesic monotone (strictly geodesic monotone). (Udrişte, 1976)

C. Udrişte gives the inequality which expresses the geodesic monotonicity in terms of df, the differential of f. T. Rapcsák [RA] states the required inequality in explicit form, using the gradient of f. However, neither of them speaks (in general) about monotone vector fields.

3 Geodesic monotone vector fields and the first variation of the length of a geodesic

In this section we make a connection between the geodesic monotone vector fields and the first variation of the lengths of geodesics. First of all we recall some preliminary definitions and results:

Definition 3.1 Let (M, ⟨ . , . ⟩) be a Riemannian manifold and ξ : [λ, ν] → M a smooth curve. A smooth deformation of the curve ξ is a smooth map

δ : [λ, ν] × [−ϵ, ϵ] → M; ϵ > 0,

with δ(t, 0) = ξ(t); ∀t ∈ [λ, ν]. Let s be the parameter which varies in [−ϵ, ϵ]. The vector field V along ξ defined by

V (t) = \frac{dδ}{ds}|_{s=0}.

is called the infinitesimal deformation induced by δ. For a fixed s ∈ [−ϵ, ϵ], the curve ξ_{s}(t) = δ(t, s) is called the deformed curve; its arclength is a smooth function

L(s) = ∫_{λ}^{ν} ∥ξ′_{s}(t)∥dt,

where ∥ . ∥ is the norm induced by ⟨ . , . ⟩. We call the first variation of the arclength of ξ corresponding to the deformation δ the number

L′(0) = \frac{dL}{ds}|_{s=0}.

Then we have the following proposition:

Proposition 3.2 Let ξ : [λ, ν] → M be a smooth curve of a Riemannian manifold, parametrized by arclength, δ a smooth deformation of ξ and V : [λ, ν] → T M the infinitesimal deformation induced by δ. Then the first variation of the length of ξ is

L′(0) = − ∫_{λ}^{ν}⟨ξ′′(t), V (t)⟩dt + ⟨ξ′ , V ⟩|_{λ}^{ν}.  (3.1)

The next theorem is the main result of this section, connecting the notion of geodesic monotonicity with the first variations of the lengths of geodesics:

Theorem 3.3 Let (M, ⟨ . , . ⟩) be a Riemannian manifold, K an open and geodesic convex subset of M and X a smooth vector field on M. Then, X is geodesic monotone (strictly geodesic monotone), if and only if for all unit speed geodesic arcs γ : [0, l] → M contained in K and all deformation δ inducing the infinitesimal variation V equal to the restriction of X to γ, the first variation of the length of γ is nonnegative (positive).

Proof. Let x, y two arbitrary, distinct points of M and γ : [0, l] → M a unit speed geodesic arc contained in K and joining x, y, such that γ(0) = x and γ(l) = y. Let δ be a deformation of γ inducing the infinitesimal variation V equal to the restriction of X to γ. Since γ is a geodesic, the first term in the variation formula (3.1) vanish. Hence we have for the first variation of the length of γ:

L′(0) = ⟨γ′, V ⟩|_{0}^{l}.

Since V is the restriction of X to γ, we have that

L′(0) = ⟨γ′(l), X_{y}⟩ − ⟨γ′(0), X_{x}⟩.  (3.2)

Using the definition of geodesic monotonicity (strict geodesic monotonicity), (3.2) proves the theorem. □

4 Closed geodesics and monotone vector fields

In this section we analyze the connection between the existence of closed geodesics and geodesic monotone vector fields. It is easy to prove the following

Theorem 4.1 Let (M, ⟨ . , . ⟩) be a Riemannian manifold which contains a closed geodesic γ. Then there are no strictly geodesic monotone vector fields on M.

Similarly we have the following

Theorem 4.2 If every geodesic of a Riemannian manifold is closed (for such manifolds see [BE]) then there are no virtually monotone vector fields on the manifold.

Since every compact Riemannian manifold contains a closed geodesic [KL], we have the following theorem:

Theorem 4.3 There is no strictly geodesic monotone vector field on a compact Riemannian manifold.

5 The geodesic monotonicity of position vector fields

In this section we define the notion of f-position vector fields of a Riemannian manifold M and analyze its geodesic monotonicity under certain topological and metrical restrictions imposed on M. By the Gauss lemma ([ON] p.127) it can be easily seen that the following definition generalize the notion of position vector fields ([ON] p.178) on complete, simply connected Riemannian manifolds of nonpositive sectional curvature. (The definition is given just for local position vector fields. However by the theorem of Hadamard ([ON] p.278) these vector fields can be defined globally on complete simply connected Riemannian manifolds of nonpositive sectional curvature.)

Definition 5.1 Let M be a complete, simply connected Riemannian manifold of nonpositive sectional curvature and o ∈ M. If x ≠ o, denote by l_{x} the distance of x from o, and by γ_{x} the unit speed radial geodesic joining o with x; γ_{x}(0) = o. Let f : [0, ∞[→ [0, ∞[ be a smooth, strictly increasing function with f(0) = 0. Then the smooth vector field P^{f}, defined by

P_{x}^{f} = \{ \begin{matrix}f(l_{x})γ′_{x}(l_{x}), if x ≠ o, \\ 0, if x = o. \end{matrix}

is called the f-position vector field of M at o.

In particular the position vector field at o of a complete, simply connected manifold M with nonpositive sectional curvature is equal to the id-position vector field of M at o, where id denotes the identity map of [0, ∞[. The local f-position vector field at a point of an arbitrary Riemannian manifold (in a normal neighbourhood) can be defined similarly. In the following definition indices i = \overline{1, 3} are considered modulo 3:

Definition 5.2 A geodesic triangle T in a Riemannian manifold M is a set formed by three segments of minimizing unit speed geodesics (called sides of the triangle)

γ_{i} : [0, l_{i}] → M; i = \overline{1, 3},

in such a way that γ_{i}(l_{i}) = γ_{i+1}(0); i = \overline{1, 3}. The endpoints of the geodesic segments are called vertices vertices of T. The angle

(−γ′_{i}(l_{i}), γ′_{i+1}(0)); i = \overline{1, 3}

is called the (interior) angle of the corresponding vertex.

Then we have the following lemma, proved in [CA] p. 259.

Lemma 5.3 Let M be a complete simply connected Riemannian manifold, with sectional curvature κ ≤ 0. Let a, b and c be three points of M. Then (by the Hadamard theorem [ON], p. 278) such points determine a unique geodesic triangle T in M with vertices a, b, c. Let α, β and γ be the angles of the vertices a, b, c, respectively, and let A, B, C be the lengths of the sides opposite the vertices a, b, c, respectively. Then

(i) A^{2} + B^{2} − 2AB cos γ ≤ C^{2} (< C^{2}, if κ < 0)

(ii) α + β + γ ≤ π (< π, if κ < 0).

Corollary 5.4 Let M be a complete simply connected Riemannian manifold with nonpositive sectional curvature. Let a, b and c be three points of M. Then (by the Hadamard theorem [ON], p. 278) such points determine a unique geodesic triangle T in M with vertices a, b, c. Let α, β and γ be the angles of the vertices a, b, c, respectively, and let A, B, C be the lengths of the sides opposite the vertices a, b, c, respectively. If f : [0, ∞[→ [0, ∞[ is a strictly increasing function, then we have:

f(C) cos β + f(B) cos γ > 0.

Proof. By (ii) of Lemma 5.3 we have that β + γ < π. Hence either β < π/2, or γ < π/2. Without losing the generality we can suppose that β ≤ γ. We consider two cases:

(1) 0 < β ≤ γ < \frac{π}{2}. In this case the inequality of the lemma is trivial.

(2) γ ≥ \frac{π}{2} and β < \frac{π}{2}. In this case we have cos γ < 0. Hence by (i) of Lemma 5.3 we have that

B^{2} < A^{2} + B^{2} − 2AB cos γ ≤ C^{2}.

Thus we get 0 < B < C, and consequently 0 ≤ f(0) < f(B) < f(C), since f is strictly increasing. Using (ii) of Lemma 5.3 and cos β > 0 we obtain that

f(C) cos β + f(B) cos γ > f(B)(cos β + cos γ) > 0. □

Theorem 5.5 Let M be a complete, simply connected Riemannian manifold with nonpositive sectional curvature and f : [0, ∞[→ [0, ∞[ a smooth, strictly increasing function with f(0) = 0. Then for all o ∈ M the f-position vector field at o is strictly geodesic monotone.

Proof. Let x ≠ y ∈ M

1. If x, y belongs to the same geodesical ray (with respect to o), then we trivially have that

⟨P_{x}^{f}, γ′(0)⟩ < ⟨P_{x}^{f}, γ′(l)⟩,

where γ : [0, l] → M is the unit speed geodesic arc joining x and y, such that γ(0) = x and γ(l) = y.

2. If x, y do not belong to the same geodesical ray the above inequality can be easily deduced by using Corollary 5.4. □ In particular the position vector field at o is monotone. However the monotonicity of this vector field is an easy consequence of Theorem 2.4, Gauss lemma ([ON] p.127), Corollary 3. [ON] p.128, and the convexity of the distance function from a point in a complete, simply connected Riemannian manifold of nonpositive sectional curvature. However we remark that an f-position vector field in general is not a gradient of a convex function.

6 Geodesic scalar derivative and geodesic monotone vector fields

In this section we shall generalize the notion of scalar derivative, and some of our results concerning the characterization of monotone operators through the scalar derivative, included in our previous papers [NE1], [NE2]. Let us start by defining the geodesic scalar derivative of a vector field:

Definition 6.1 Let (M, ⟨ . , . ⟩) be a Riemannian manifold, K ⊂ M a geodesic convex open set and X ∈ Sec(T K) a vector field on K. Then the lower geodesic scalar derivative (upper geodesic scalar derivative) of X is the function

\underline{X}^{#} : K → R; \underline{X}^{#}(x) = lim inf_{{t↘0 \atop γ∈Γ}} \frac{g(X_{γ(t)}, γ̇(t)) − g(X_{x}, γ̇(0))}{t}

(\overline{X}^{#} : K → R; \overline{X}^{#}(x) = lim sup_{{t↘0 \atop γ∈Γ}} \frac{g(X_{γ(t)}, γ̇(t)) − g(X_{x}, γ̇(0))}{t}),

where Γ denotes the family of unit speed geodesic arcs γ : [0, l] → M starting from x (i.e γ(0) = x), and contained in K. If x_{0} ∈ K and \underline{X}^{#}(x_{0}) = \overline{X}^{#}(x_{0}) =: X^{#}(x_{0}), then X is called geodesic scalar differentiable in x_{0} and X^{#}(x_{0}) is called the geodesic scalar derivative of X in x_{0}. If X is geodesic scalar differentiable in every x ∈ K then X is called geodesic scalar differentiable, and the function X^{#} : x ↦ X^{#}(x) is called the geodesic scalar derivative of X.

The following theorem is a local characterization of geodesic monotone vector fields.

Theorem 6.2 Let (M, ⟨ . , . ⟩) be a Riemannian manifold, K ⊂ M a geodesic convex open set and X ∈ Sec(T K) a vector field on K. Then we have the following assertions:

(i) X (−X) is geodesic monotone if and only if \underline{X}^{#}(x) ≥ 0 (\overline{X}^{#}(x) ≤ 0), for all x ∈ K.

(ii) X is trivially monotone if and only if X is scalar differentiable and X^{#}(x) = 0, for all x ∈ K

Proof.

(i) First we suppose that \underline{X}^{#}(x) ≥ 0 for every x ∈ K and prove that X is geodesic monotone Let a and b two arbitrary distinct points of K and γ : [0, l] → M a unit speed geodesic arc joining a and b (γ(0) = a, γ(l) = b) contained in K. Let x = γ(t) be an arbitrary point of the geodesic arc.

Let ϵ > 0 be an arbitrary but fixed positive number. Since \underline{X}^{#}(x) ≥ 0, there is a δ(t) > 0 such that for every s ∈ I_{t} =]t − δ(t), t + δ(t)[ we have that

\frac{⟨X_{γ(s)}, γ′(s)⟩ − ⟨X_{x}, γ′(t)⟩}{s − t} > −\frac{ϵ}{2l}.  (6.1)

(The geodesic arc can be continued to an open one contained in K and δ(t) choosed sufficiently small such that γ(s) to be defined.) But {I_{t} : t ∈ [0, l]} is an open covering of the compact set [0, l]. Hence [0, l] ⊂ I_{t_{1}} ∪ I_{t_{2}} ∪ … ∪ I_{t_{m−1}} for some positive integer m and some points t_{1} < t_{2} < … < t_{m−1} of [0, l]. This yields 0 =: t_{0} ∈ I_{t_{1}} and l =: t_{m} ∈ I_{t_{m−1}}. Obviously we can choose the intervals \{I_{i} : i = \overline{1, m − 1}\} such that no interval to be contained in another. Let ξ_{i} ∈ I_{t_{i−1}} ∩ I_{t_{i}}∩]t_{i−1}, t_{i}[ for i = \overline{1, m}. Then using (6.1) we have that

⟨X_{γ(ξ_{i})}, γ′(ξ_{i})⟩ − ⟨X_{γ(t_{i−1})}, γ′(t_{i−1})⟩ > −\frac{ϵ}{l}(ξ_{i} − t_{i−1}),  (6.2)

and

⟨X_{γ(t_{i})}, γ′(t_{i})⟩ − ⟨X_{γ(ξ_{i})}, γ′(ξ_{i})⟩ > −\frac{ϵ}{l}(t_{i} − ξ_{i}),  (6.3)

for i = \overline{1, m}. Summing inequalities (6.2) and (6.3) we obtain

⟨X_{γ(t_{i})}, γ′(t_{i})⟩ − ⟨X_{γ(t_{i−1})}, γ′(t_{i−1})⟩ > −\frac{ϵ}{l}(t_{i} − t_{i−1}),  (6.4)

for i = \overline{1, m}. Summing inequalities (6.4) for i = \overline{1, m} we get

⟨X_{γ(l)}, γ′(l)⟩ − ⟨X_{γ(0)}, γ′(0)⟩ > −ϵ.

Since ϵ is an arbitrary positive number we have that

⟨X_{γ(l)}, γ′(l)⟩ − ⟨X_{γ(0)}, γ′(0)⟩ ≥ 0.

Hence for arbitrary two distinct points a and b of K and arbitrary geodesic arc joining a and b contained in K, we have that

⟨X_{b}, γ′(l)⟩ ≥ ⟨X_{a}, γ′(0)⟩.

Thus X is geodesic monotone. Now suppose that X is geodesic monotone. Then by the definitions of geodesic monotonicity and geodesic scalar derivative, we have that \underline{X}^{#}(x) ≥ 0, for all x ∈ K. The statement of (i) for −X is obtained by using the identity

\underline{(−X)}^{#} = −\overline{X}^{#}.

(ii) We have that X is trivially geodesic monotone, if and only if X and −X is geodesic monotone. Hence X is trivially geodesic monotone, if and only if 0 ≤ \underline{X}^{#}(x) ≤ \overline{X}^{#}(x) ≤ 0, for all x ∈ K. Thus X is trivially geodesic monotone if and only if it is scalar differentiable and X^{#}(x) = 0 for every x ∈ K. Similarly to Theorem 6.2 it can be proved the following theorem:

Theorem 6.3 Let (M, ⟨ . , . ⟩) be a Riemannian manifold, K ⊂ M a geodesic convex open set and X ∈ Sec(T K) a vector field on K. If \underline{X}^{#}(x) > 0 for every x ∈ K, then X is strictly geodesic monotone.

For the smooth case we have the following corollary of Theorem 6.2:

Corollary 6.4 Let M be a Riemannian manifold, ∇ the Levi-Civitá connection of M, K ⊂ M a geodesic convex set and X ∈ Sec(T K) a smooth vector field on K. For x ∈ K define ϕ(x) : T_{x}(M) → T_{x}(M) by ϕ(x)(v) = ∇_{v}X for v ∈ T_{x}(M), where T_{x}(M) is the tangent space of M in x. Then we have the following assertions:

(i) \underline{X}^{#}(x) = inf_{{v∈Tx(M) \atop ∥v∥=1}}⟨ϕ(x)(v), v⟩

(ii) X is geodesic monotone if and only if ϕ(x) is positive semidefinite, for every x in K

(iii) X is trivially monotone if and only if ϕ(x) is antisymmetric, for every x ∈ K

(iv) If ϕ(x) is positive definite, for every x ∈ K, then X is strictly geodesic monotone.

Proof. Let x be an arbitrary point of K. Then we have aaa

\underline{X}^{#}(x) = inf_{γ∈Γ} lim_{t↘0} \frac{⟨X_{γ(t)}, γ′(t)⟩ − ⟨X_{x}, γ′(0)⟩}{t}.  (6.5)

Equation (6.5) can be rewritten as

\underline{X}^{#}(x) = inf_{γ∈Γ} \frac{d}{dt}|_{t=0^{+}}⟨X_{γ(t)}, γ′(t)⟩.  (6.6)

Using that ∇ is the Levi-Civitá connection of M and γ is geodesic, (6.6) becomes

\underline{X}^{#}(x) = inf_{γ∈Γ}⟨∇_{γ′(0)}X|_{x}, γ′(0)⟩.  (6.7)

Since for every v ∈ T_{x}(M) there is γ ∈ Γ such that γ′(0) = v , from (6.7) we obtain that

\underline{X}^{#}(x) = inf_{{v∈Tx(M) \atop ∥v∥=1}}⟨ϕ(x)(v), v⟩,  (6.8)

where ∥ . ∥ is the norm generated by the metrical tensor of M. Using (i) From (6.8) and Theorems 6.2 and 6.3 we obtain the assertions (ii), (iii) and (iv) of the corollary. □ Using Theorem 6.2 and Corollary 6.4 we obtain:

Theorem 6.5 Let (M, ⟨ . , . ⟩) be a Riemannian manifold, K ⊂ M a geodesic convex open set and X ∈ Sec(T K) a smooth vector field on K. Then the following assertions are equivalent:

(i) X is trivially geodesic monotone,

(ii) X is a Killing vector field

Proof. By Corollary 6.4 (iii) we have that X is trivially geodesic monotone if and only if A_{Y} is antisymmetric, where A_{Y} is the endomorphism of smooth vector fields defined by A_{Y} (X) = ∇_{Y} X. But this is exactly a necessary and sufficient condition for X to be a Killing vector field.

7 Geodesic monotone vector fields on S^{n}

Denote by ⟨ . , . ⟩ the canonical scalar product of R^{n+1}. Consider S^{n}, the standard unit sphere, to be a Riemannian manifold, with the metric induced from R^{n+1} (we shall denote its metric by ⟨ . , . ⟩ too). We start this section by remarking that there are no virtually monotone vector fields on the whole sphere. This is an obvious consequence of Theorem 4.1, since every geodesic of the sphere is closed. Hence we can seek for virtually monotone vector fields just on geodesic convex proper subsets of the sphere. Denote by {g \atop y→x} the convergence of y to x along geodesics. Then we have the following theorem:

Theorem 7.1 Let K be an open and geodesic convex subset of S^{n}, X a vector field on K and x an arbitrary point of K. Then we have the following formula for the lower geodesic scalar derivative of X in x:

\underline{X}^{#}(x) = lim inf_{{g \atop y→x}} \frac{⟨X_{y} − X_{x}, y − x⟩}{∥y − x∥^{2}}.

Proof. Every unit speed geodesic of the n-dimensional sphere S^{n} starting from x ∈ S^{n} (γ(0) = x) has equation:

γ(t) = cos t x + sin t e,  (7.1)

where e = γ′(0) ∈ T_{x}(S^{n}) is the tangent unit vector of γ in the starting point. We want to find the equation of the unit speed geodesic, denoted by the same letter γ, starting from x ∈ S^{n} and passing through y ∈ S^{n}. Obviously this is equivalent to finding the tangent unit vector e of γ in the starting point. This can be calculated in the following way: First we find the parameter l corresponding to y and after that we can compute the unit tangent vector of γ e = γ′(0) using (7.1). Having in mind that x and y are unit vectors of T_{0}(R^{n+1}), the tangent space of R^{n+1} in the origin, we have that

cos l = ⟨x, y⟩.  (7.2)

Thus we get

e = ±\frac{y − ⟨x, y⟩x}{\sqrt{1 − ⟨x, y⟩^{2}}} .  (7.3)

¿From (7.1), (7.2) and (7.3) we have that

γ′(l) = ∓\sqrt{1 − ⟨x, y⟩^{2}} x ± ⟨x, y⟩\frac{y − ⟨x, y⟩x}{\sqrt{1 − ⟨x, y⟩^{2}}} .  (7.4)

If Γ has the same meaning as in the previous section then we have

\underline{X}^{#}(x) = lim inf_{{l↘0 \atop γ∈Γ}} \frac{⟨X_{γ(l)}, γ′(l)⟩ − ⟨X_{γ(0)}, γ′(0)⟩}{l}.  (7.5)

Since lim_{l→0}(sin l)/l = 1 (7.5) becomes

\underline{X}^{#}(x) = lim inf_{{l↘0 \atop γ∈Γ}} \frac{⟨X_{γ(l)}, γ′(l)⟩ − ⟨X_{γ(0)}, γ′(0)⟩}{sin l}.  (7.6)

Inserting γ(0) = x, γ(l) = y, (7.4), (7.3), ⟨X_{x}, x⟩ = ⟨X_{y}, y⟩ = 0 and sin l = ±\sqrt{1 − ⟨x, y⟩^{2}} in (7.6) we obtain

\underline{X}^{#}(x) = lim inf_{{g \atop y→x}} \frac{⟨X_{y} − X_{x}, y − x⟩}{1 − ⟨x, y⟩^{2}}.  (7.7)

Since 1 − ⟨x, y⟩^{2} = (1 − ⟨x, y⟩)(1 + ⟨x, y⟩) and 1 + ⟨x, y⟩ → 2, when {g \atop y→x}, (7.7) becomes

\underline{X}^{#}(x) = lim inf_{{g \atop y→x}} \frac{⟨X_{y} − X_{x}, y − x⟩}{2 − 2⟨x, y⟩}.  (7.8)

But since ∥x∥ = ∥y∥ = 1 we have that 2 − 2⟨x, y⟩ = ∥y − x∥^{2}, which inserted in (7.8) gives the required equality of the theorem. □

For the smooth case we have the following Corollary of Theorem 7.1:

Corollary 7.2 If X = f|_{K}, where f : R^{n+1} → R^{n+1} is a smooth operator satisfying ⟨f(x), x⟩ = 0 when ∥x∥ = 1, then

\underline{X}^{#}(x) = inf_{{∥h∥=1, \atop ⟨x, h⟩=0}}⟨df(x)(h), h⟩,

where df(x) is the Frechét differential of f in x.

Proof. Suppose that y → x along the arbitrary but fixed unit speed geodesic γ starting from x (γ(0) = x). This means that y = γ(t), where t ↘ 0. Hence using X = f|_{K} we have that

\frac{⟨X_{y} − X_{x}, y − x⟩}{∥y − x∥^{2}} =

⟨ \frac{f(γ(t)) − f(γ(0))}{t} : ∥ \frac{γ(t) − γ(0)}{t} ∥ , \frac{γ(t) − γ(0)}{t} : ∥ \frac{γ(t) − γ(0)}{t} ∥ ⟩

(7.9)

If y → x along γ i.e t ↘ 0 then using (d/dt)|_{t=0}f(γ(t)) = df(x)(γ′(0)), and that γ is of unit speed we obtain from (7.9):

lim_{{y→x \atop y∈Imγ}} \frac{⟨X_{y} − X_{x}, y − x⟩}{∥y − x∥^{2}} = ⟨df(x)(γ′(0)), γ′(0)⟩,  (7.10)

where we have denoted by Imγ the image of γ. Equation (7.10) holds for every γ, hence by definition of \underline{X}^{#}(x) we get

\underline{X}^{#}(x) = inf_{γ∈Γ}⟨df(x)(γ′(0)), γ′(0)⟩.  (7.11)

Since for every h ∈ T_{x}(S^{n}), ∥h∥ = 1 there is a γ ∈ Γ such that γ′(0) = h, (7.11) becomes the required equality of the corollary. □

Remarks 7.3 We shall use the same notations as above. Then we have that 1. Similarly to Theorem 7.1 it can be proved that X is geodesic monotone (strictly geodesic monotone) if and only if

⟨X_{y} − X_{x}, y − x⟩ ≥ 0 (⟨X_{y} − X_{x}, y − x⟩ > 0),

for arbitrary two distinct points x and y of K. If X is smooth, using Theorem 6.2 and Corollary 7.2 we have that X is geodesic monotone if and only if

⟨df(x)(h), h⟩ ≥ 0,

for all x ∈ K and all h ∈ R^{n+1} − {0} with ⟨x, h⟩ = 0. Similarly using Theorem 6.3 and Corollary 7.2 if X is smooth and

⟨df(x)(h), h⟩ > 0,

for all x ∈ K and all h ∈ R^{n+1} − {0} with ⟨x, h⟩ = 0, then X is strictly geodesic monotone.

2. If f = A, where A is an antisymmetric operator, then it follows easily that X is trivially geodesic monotone.

In the following theorem we shall give a method for computing the scalar derivative of a vector field on the sphere.

Theorem 7.4 Let X be a vector field on the sphere, such that X = f|_{S^{n}}, where f : R^{n+1} → R^{n+1} is a smooth operator satisfying ⟨f(x), x⟩ = 0, for all x ∈ R^{n+1} with ∥x∥ = 1. Let x ∈ S^{n} and B(x) an orthogonal matrix having in the first column the components of x relative to the canonical basis of R^{n+1} and

C(x) = B^{T} (x)J_{s}f(x)B(x),

where T denotes transposition and

J_{s}f(x) = (Jf(x) + Jf(x)^{T})/2

is the symmetrizant of the Jacobi matrix of f in x denoted by Jf(x). Let C^{−}(x) be the n × n matrix obtained from C(x) by deleting the first row and first column. Then \underline{X}^{#}(x) is the smallest eigenvalue of C^{−}(x).

Before proving Theorem 7.4 we shall state a corollary of it.

Corollary 7.5 If K ⊂ S^{n} is open and geodesic convex and X is a vector field on K such that C^{−}(x) is positive definite for all x ∈ K then X is strictly geodesic monotone.

The Proof of Corollary 7.5 is an easy consequence of Theorems 6.3 and 7.4.

Proof of Theorem 7.4. Corollary 7.2 implies that

\underline{X}^{#}(x) = inf_{{∥h∥=1 \atop ⟨x, h⟩=0}}⟨J_{s}f(x)h, h⟩.

Using the conditional extreme value theorem we obtain

\underline{X}^{#}(x) = inf\{λ : J_{s}f(x)h = λh + μx, for some μ ∈ R, and h ∈ R^{n+1}
with ∥h∥ = 1 and ⟨x, h⟩ = 0}.

(7.12)

Consider an orthonormal basis of R^{n+1} with first vector x. Then in the new base we have: h = (0, h_{2} …, h_{n+1}) and the matrix of d_{s}f(x) = (df(x) + df(x)^{T})/2 is C(x). If C(x) = (c_{ij}(x))_{1≤i,j≤n+1}, then equation J_{s}f(x)h = λh + μx in the new base is equivalent to

μ = c_{12}h_{2} + … + c_{1,n+1}h_{n+1}

and

C^{−}(x)h^{−} = λh^{−},  (7.13)

where h^{−} = (h_{2}, …, h_{n+1}). ¿From (7.12) and (7.13) follows immediately the statement of the theorem. □

Example 7.6 Let f : R^{3} → R^{3}, f(x) = (x_{1}^{2} − 1, x_{1}x_{2}, x_{1}x_{3}), for x = (x_{1}, x_{2}, x_{3}). Since (f(z), z) = 0 for all z ∈ S^{2} there is a smooth vector field X on S^{2} such that X = f|_{S^{2}}. Hence X_{z} = (z_{1}^{2} − 1, z_{1}z_{2}, z_{3}), for arbitrary point z = (z_{1}, z_{2}, z_{3}) of S^{2}. Using Theorem 7.4 by a lengthy but easy computation we obtain \underline{X}^{#}(z) = z_{1}. Using Theorem 6.3 the vector field X is strictly geodesic monotone on the

\{ \begin{matrix}z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 1, \\ z_{1} > 0 \end{matrix}

half sphere. □

We remark that similar results to those of this section can be obtained for any constant curvature manifold.

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[UD2] C. Udrişte, Continuity of Convex Functions on Riemannian Manifolds, Bulletine Mathematique de Roumanie 21 (1977) 215-218 Laboratory of Operation Research and Decision Systems, Computer and Automation Institute, Hungarian Academy of Sciences, H-1518 Budapest, P.O.Box 63 Hungary E-mail address: snemeth@oplab.sztaki.hu