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Local midpoints on smooth manifolds
Differential Geometry and its Applications 39 (2015) 129–146
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Differential Geometry and its Applications www.elsevier.com/locate/difgeo
Local midpoints on smooth manifolds Sejong Kim a , Jimmie Lawson b,∗ a b
Department of Mathematics, Chungbuk National University, Cheongju, 361-763, Republic of Korea Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
a r t i c l e
i n f o
Article history: Received 6 March 2014 Available online xxxx Communicated by B. Ørsted MSC: 53B15 53C35 53B20
a b s t r a c t In this paper we consider three methods for obtaining midpoints, primarily midpoints of geodesics of sprays, but also midpoints of symmetry (in symmetric spaces), and metric midpoints (in Riemannian manifolds). We derive general conditions under which these approaches yield the same result. We also derive a version of the Lie–Trotter formula based on the midpoint operation and use it to show that continuous maps preserving (local) midpoints are smooth. © 2015 Elsevier B.V. All rights reserved.
Keywords: Midpoint Spray Lie–Trotter formula Smooth manifold Symmetric space Riemannian manifold
1. Introduction There are various natural notions of a “midpoint” that present themselves. The first is a geodesic approach: given a notion of a geodesic α that is uniquely minimal between a = α(t0 ) and b = α(t1 ), one may take as midpoint α((t0 + t1 )/2). The second is a metric approach: given a, b in a metric space, we may take as midpoint a point m such that d(a, m) = d(m, b) = (1/2)d(a, b), provided such an m uniquely exists. The third approach is what we might call a midpoint of symmetry: given a set X endowed with a geometric structure that includes a point reflection Sa : X → X through each point a, we may define a midpoint m of a and b to be one satisfying Sm (a) = b and vice-versa, again provided such an m uniquely exists. The principal approach of this paper is to consider smooth manifolds equipped with sprays, a variant of connections. The sprays give rise to geodesics, which in small enough neighborhoods are unique. We review and slightly rework this machinery in Section 2, since it is crucial to what follows. In Section 3 we define * Corresponding author. E-mail addresses: [email protected] (S. Kim), [email protected] (J. Lawson). http://dx.doi.org/10.1016/j.difgeo.2015.01.009 0926-2245/© 2015 Elsevier B.V. All rights reserved.
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midpoints locally via the geodesic approach as in the preceding paragraph. Indeed we can define weighted means a#t b for 0 ≤ t ≤ 1, with the midpoint corresponding to t = 1/2. We further show in Section 3 that the local midpoint operation satisfies a variant Lie–Trotter product formula based on the local binary midpoint operation. The standard Lie–Trotter product formula n eA+B = lim eA/n eB/n n→∞
first formulated by Sophus Lie for square matrices A and B is of great utility and generalizes to various settings such as semigroups of (unbounded) operators [7] and Lie theory. For an elementary derivation and basic applications of the formula in the setting of Lie theory, we refer the reader to [1]. The famous fifth problem of Hilbert asked whether a locally euclidean topological group must have a differentiable multiplication, i.e., must be a Lie group. More generally the problem asked for general conditions for a continuous function to be smooth. As an application of our results in the fourth section we apply our Lie–Trotter formula to show this is true (in the category of smooth manifolds equipped with sprays) for continuous functions preserving the (local) midpoint operation. In Section 5 we consider midpoints of symmetry in the context of smooth symmetric spaces modeled on general Banach spaces. Using a crucial result of K.-H. Neeb [6] that a smooth symmetric space uniquely determines a canonical spray, we establish that locally the midpoints of symmetry arising from the symmetric space structure agree with the geodesic midpoints of the associated spray. In Section 6 we turn to Riemannian manifolds. Any Riemannian manifold X has a unique connection associated with the Riemannian metric g, called the Levi-Civita connection or the canonical connection. This is typically used to define the geodesics of the Riemannian manifold, although in [2, Section VII.7] Lang has defined them alternatively (and equivalently) from the canonical spray associated to a Riemannian structure. We show that locally the metric midpoints with respect to the distance function determined by g agree with the geodesic midpoints of the associated canonical spray associated with the Levi-Civita connection. We close this section with the illustrative example of the symmetric space of all positive definite matrices, where our three approaches to defining midpoints all converge. In the final section we indicate briefly how our results generalize to the setting of Finsler manifolds. 2. Sprays and exponential maps We work in the category of smooth manifolds modeled on Banach spaces and smooth maps, where smooth means C ∞ . The tangent bundle of a smooth manifold X is denoted π : TX → X. A vector field on X means a smooth map ξ : X → TX such that ξ(x) lies in the tangent space Tx X for each x ∈ X, i.e., π ◦ ξ = idX . An integral curve for ξ with initial condition x0 ∈ X is a smooth curve α : J → X mapping an open interval J ⊆ R containing 0 into X such that for all t ∈ J α (t) = ξ(α(t)), and α(0) = x0 .
(2.1)
From Theorem 2.2 in [2, Chapter IV.2], we obtain the following result. Lemma 2.1. Let ξ : X → TX be a smooth vector field. Then there exists a smooth map Φ : U → X, where U is an open subset of R × X such that {0} × X ⊆ U , satisfying the following. (1) Φ(0, x) = x.
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(2) Φ(s, Φ(t, x)) = Φ(s + t, x) provided Φ(t, x) is defined and either the left-hand or right-hand side is defined. (3) Let J(x) = (t− (x), t+ (x)) be the largest time interval for x containing 0 such that (t− (x), t+ (x)) ×{x} ⊆ U = dom(Φ). Then ϕx (t) on J(x) defined by ϕx (t) = Φ(t, x) is a maximal integral curve, in particular ϕx (t) = ξ(ϕx (t)). Furthermore, the solution ϕx of (2.1) is unique on any interval on which it is defined with ϕx (0) = x. The smooth map Φ in Lemma 2.1 is called a local flow, item (1) the identity property, and item (2) the semigroup property. Definition 2.2. A second-order vector field is a vector field F : TX → T (TX ) satisfying T (π) ◦ F = idTX . Applying Lemma 2.1 to a second-order vector field F , we obtain a local flow ΦF : UF → TX , where UF is an open subset of R × TX containing {0} × TX , such that for any vector v ∈ TX (1) ΦF (0, v) = v, (2) ΦF (s, ΦF (t, v)) = ΦF (s + t, v) provided ΦF (t, v) is defined and either the left-hand or right-hand side is defined. The condition for a vector field to be a second-order vector field can be stated alternatively in terms of the flow of the vector field, see [2, Chapter IV.3]. Proposition 2.3. Let F : TX → T (TX ) be a vector field. Then F is a second-order vector field if and only if the local flow ΦF on the tangent space TX satisfies (π ◦ ΦF ) = ΦF , more specifically, ∂ (π ◦ ΦF )(t, x) = ΦF (t, x). ∂t Definition 2.4. Let s ∈ R and s∗ : TX → TX denote the scalar multiplication by s in each tangent space. A second-order vector field F on TX is called a spray if for all s ∈ R and v ∈ TX F (sv) = T (s∗ )(sF (v)).
(2.2)
In [2, Chapter IV.4] S. Lang presents alternative equivalent conditions for defining a spray in terms of the integral curves of the second-order vector field F . We translate these to equivalent conditions on the local flow. Proposition 2.5. Let ΦF be the local flow of the second-order vector field F with initial condition v. Then for v ∈ TX , each of the following is equivalent to the condition (2.2): (1) A pair (t, sv) is in the domain of ΦF if and only if (st, v) is in the domain of ΦF , and then ΦF (t, sv) = sΦF (st, v).
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(2) A pair (st, v) is in the domain of ΦF if and only if (s, tv) is in the domain of ΦF , and then πΦF (s, tv) = πΦF (st, v). (3) A pair (t, v) is in the domain of ΦF if and only if (1, tv) is in the domain of ΦF , and then πΦF (t, v) = πΦF (1, tv). Lemma 2.6. Given a spray F : TX → TTX , a C 1 -map α : J → X from some interval J ⊆ R, and v ∈ TX , the following are equivalent: (1) for all t ∈ J, α(t) = πΦF (t, v); (2) α = π ◦ ϕv |J , where ϕv (t) = ΦF (t, v) for t ∈ J(v); (3) for all t ∈ J, α (t) = ΦF (t, v). Under these equivalent hypotheses α is the composition of smooth functions, hence smooth. Proof. One sees directly that (1) and (2) are equivalent. That (1) implies (3) follows from Proposition 2.3. Assume (3) holds. Then for t ∈ J, πΦF (t, v) = π(α (t)) = α(t), so (1) is satisfied. The last assertion follows from (2). 2 Definition 2.7. A geodesic of a spray F is a (necessarily smooth) map α : J → X from some interval J ⊆ R satisfying any of the equivalent conditions of Lemma 2.6. Remark 2.8. For a geodesic α, a reparametrization β(t) = α(bt + c) is again a geodesic since β(t) = α(bt + c) = πΦF (bt + c, v) = πΦF (bt, ΦF (c, v)) = πΦF (t, bΦF (c, v)). We now recall how the local flow ΦF of a spray F gives rise to its associated exponential map. Let D = {v ∈ TX : [0, 1] × {v} ⊆ dom(ΦF )}. We know from Corollary 2.7 in [2, Chapter IV.2] that D is an open set in TX , and the map v → ΦF (1, v) is a smooth map of D into TX . We define the exponential map exp : D → X, exp(v) = πΦF (1, v) and call D the domain of the exponential map associated with the spray F . Note that the map exp is smooth. If x ∈ X and 0x denotes the zero vector in Tx X, then we obtain exp(0x ) = x. Indeed, exp(0x ) = πΦF (1, 0x ) = πΦF (1, 0 · 0x ) = πΦF (0, 0x ) = π(0x ) = x. Thus the exponential map is defined and coincides with π on the zero section, and so induces an diffeomorphism of the cross section onto X. We denote the restriction of exp to the tangent space Tx X by expx : expx : D ∩ Tx X → X.
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Remark 2.9. (1) From the preceding we can define expx (tv) for v in the domain of expx and t ∈ dom(ϕv ), in particular for 0 ≤ t ≤ 1, as expx (tv) = πΦF (1, tv) = πΦF (t, v) = πϕv (t). It follows for v ∈ dom(expx ) that t → expx (tv) is a geodesic on an open interval containing [0, 1]. Furthermore, we have expx (s(tv)) = πΦF (s, tv) = πΦF (st, v) = expx ((st)v) by Proposition 2.5(2) for (st, v) in the domain of ΦF , in particular for 0 ≤ s ≤ 1. (2) For v in the domain of expx , define α(t) = expx (t) and β(t) = α(1 − t) for 0 ≤ t ≤ 1. Then β(t) = expx (1 − t) = πΦF (1 − t, v) = πΦF (−t, ΦF (1, v)) = πΦF (t, −ΦF (1, v)), where the third equality follows from the fact ΦF is a local flow and the fourth from Proposition 2.5(2). It follows from this equation that β is a geodesic. We recall two important facts about the exponential function, first Theorem 4.1 of [2, Section IV.4]. Theorem 2.10. The derivative of the exponential map expx at 0x is the identity, and hence it is a local diffeomorphism from Tx X into X on a neighborhood of 0x . The second fact follows from Proposition 4.2 of [2, Section IV.4]. Proposition 2.11. For any v ∈ Tx X at which expx is defined, the map αv (t) = expx (tv) is a geodesic. Conversely if α : J → X is a geodesic with α(0) = x and α (0) = v ∈ Tx X, then α(t) = expx (tv) for all t ∈ J. For > 0 we denote by B() the open ball of elements v ∈ E, the Banach space on which X is modeled, with v < . Let x0 ∈ X. From the definition of the tangent space and the resulting local triviality of the tangent bundle, a chart defined on a neighborhood U of x0 induces a commuting triangle π −1 (U ) π
U ×E πU
U where the horizontal arrow is a diffeomorphism and a vector space isomorphism on the fibers and πU is projection into the first coordinate. It follows that if we work only in the neighborhood U , we can take the tangent bundle TU to be U × E with bundle map first projection. From the results of [2, Chapter VIII.5], we can deduce for an arbitrary point x0 ∈ X the existence of a well-behaved neighborhood W in the sense of the following theorem. Theorem 2.12. (1) Given x0 ∈ X, there exists an open set V ⊆ TX containing 0x0 such that G(v) = (πv, exp v) is a diffeomorphism from V onto some open subset of X × X containing (x0 , x0 ). We may pick V such that for some > 0 in some local trivialization of the tangent bundle V = U0 × B() ⊆ U0 × E = TU 0 .
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(2) There further exists an open neighborhood W of x0 , W ⊆ U0 , such that G(V ) ⊇ W ×W and the following additional properties are satisfied: (i) Any two points x, y ∈ W are joined by a unique geodesic t → expx (tv), 0 ≤ t ≤ 1 lying in U0 , where expx (v) = y. This geodesic depends smoothly on the pair (x, y) in the sense that the correspondence v ↔ G(v) = (πv, exp v) = (x, expx v) = (x, y) is a C ∞ -diffeomorphism between an open subset of V and W × W . (ii) For each x ∈ W the exponential expx maps the open subset {x} × B() of Tx X diffeomorphically onto an open set U (x) containing W and contained in U0 . We call the pair (W, V ) a normal neighborhood pair of x0 in X and B() a normal ball around 0x0 in Tx0 X. Remark 2.13. The geodesic α(t) = expx (tv), 0 ≤ t ≤ 1, of Theorem 2.12 is unique in the sense that it is the only geodesic map from [0, 1] into U0 such that α(0) = x and α(1) = y, where y = expx (v). 3. Means and the Lie–Trotter formula Let X be a smooth manifold with spray F . We define locally the weighted means of any pair (x, y), call the equally weighted mean the midpoint, and derive the Lie–Trotter formula with respect to this local midpoint operation. Let x0 ∈ X and let (W, V ) be a normal neighborhood pair of x0 , where V = U0 ×B(), as in Theorem 2.12. From Theorem 2.12 any two points x, y ∈ W are joined by a unique geodesic in U0 , namely t → expx (tv) with x = expx (0x ) = expx (0 · v) and y = expx (v). Unfortunately, for some s ∈ [0, 1] the image expx (sv) may lay outside the normal neighborhood W . So we consider the set DW = {(x, y) ∈ W × W : expx (tv) ∈ W for all 0 ≤ t ≤ 1, where expx (v) = y}. Lemma 3.1. The set DW is open. Proof. For any pair (x, y) ∈ DW , we define a map α : [0, 1] × W × W → U0 by α(t, x, y) = expx (tv), where 0 ≤ t ≤ 1. From the definition of DW we have that α ([0, 1] × {x} × {y}) ⊆ W . Furthermore, the map α is continuous on [0, 1] × W × W by Theorem 2.12 and the continuity of the flow Φ. By the compactness of the factors of [0, 1] × {x} × {y}, there are open sets O1 in W containing {x} and O2 in W containing {y} such that α ([0, 1] × O1 × O2 ) ⊆ W . This means that O1 ×O2 is an open set in DW containing {x} ×{y}. 2 For any pair (x, y) ∈ DW we can naturally define the weighted mean of x and y. Definition 3.2. The t-weighted mean of a pair (x, y) ∈ DW is given by x#t y = expx (tv),
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where 0 ≤ t ≤ 1, expx (v) = y, and t → expx (tv) is the unique geodesic in W from x to y. We write x#y for x# 12 y and call it the midpoint of x and y. Remark 3.3. We note that in a neighborhood of (t, x, y) for (x, y) ∈ DW , we have x#t y = expx (tG−1 (x, y)) = πΦF (1, tG−1 (x, y)), which is a composition of smooth functions (see Theorem 2.12(2)(i)) where defined, hence smooth. Remark 3.4. From Definition 3.2 we see that x# expx (tv) = expx ((t/2)v) for any t ∈ [0, 1]. Indeed, for u = tv, expx (u) ∈ W ; moreover, (x, expx (u)) ∈ DW since expx (su) = expx (stv) ∈ W for 0 ≤ s ≤ 1. Then x# expx (u) = expx
1 u 2
= expx
t v . 2
Similarly we can define the midpoint y#x for a pair (y, x) ∈ DW by y#x := expy
1 w , 2
where x = expy (w). Proposition 3.5. For any pair (x, y) ∈ DW and 0 ≤ t ≤ 1, x#t y = y#1−t x. In particular, x#y = y#x. Proof. Let α(t) = expx (tv), where y = expx (v), be a geodesic from x to y lying entirely in W . By Remark 2.9(2) β(t) = α(1−t) is a geodesic from y to x, which by definition also lies in W . By Theorem 2.12(2)(i) β(t) = expy (tu) for some u for 0 ≤ t ≤ 1. Thus y#1−t x = expy ((1 − t)u) = β(1 − t) = α(t) = expx (tv) = x#t y.
2
In the next sequence of results we show that the t-weighted mean map α(t) = expx (tv) is uniquely determined by its midpoint preservation property. Lemma 3.6. For any pair (x, y) ∈ DW with y = expx (v), the map α(t) = expx (tv), 0 ≤ t ≤ 1, is midpointpreserving from [0, 1] into W . Proof. Let 0 ≤ t1 < t2 ≤ 1, and let s = t2 − t1 . Set w = sΦF (t1 , v), x1 = π(w), and x2 = expx1 (w). We note that x1 = π(w) = π(sΦF (t1 , v)) = π(ΦF (t1 , v)) = expx (t1 v) and x2 = expx1 (w) = πΦF (1, w) = πΦF (1, sΦF (t1 , v)) = πΦF (s + t1 , v) = expx (t2 v). The following calculation now establishes midpoint preservation:
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expx
t 1 + t2 2
s s v = expx t1 + = πΦF t1 + , v 2 2 = πΦF (s/2, ΦF (t1 , v)) = πΦF (1/2, sΦF (t1 , v)) 1 w = πΦF (1/2, w) = expx1 2 2
= x1 #x2 = expx (t1 v)# expx (t2 v). The following corollary is a restatement of the previous lemma.
Corollary 3.7. For (x, y) ∈ DW , s, t ∈ [0, 1], (x#t y)#(x#s y) = x#r y, where r = (s + t)/2. Proposition 3.8. For any pair (x, y) ∈ DW , the map α(t) = expx (tv), 0 ≤ t ≤ 1, with expx (v) = y is the unique midpoint-preserving continuous map from [0, 1] into W with α(0) = x and α(1) = y. Proof. Let β : [0, 1] → W be continuous, midpoint-preserving, and satisfy β(0) = x, β(1) = y. Then by midpoint preservation β(1/2) = β
0+1 2
= β(0)#β(1) = x#y.
Thus (β(0), β(1/2)) = (x, x#y), (β(1/2), β(1)) = (x#y, y) ∈ DW . From Corollary 3.7 1 1 0+ = β(0)#β(1/2) = (x#0 y)#(x#y) = x# 14 y. β(1/4) = β 2 2 By a similar argument on the interval [1/2, 1], we conclude β(3/4) = x# 34 y. Inductively bisecting intervals and applying the same argument, we conclude that β(r) = x#r y for all dyadic rationals r ∈ [0, 1]. Thus β agrees with expx (rv) = x#r y for all dyadic rationals r and by continuity for all 0 ≤ t ≤ 1. 2 We establish some useful identities involving the weighted mean. Proposition 3.9. For any pair (x, y) ∈ DW , and r, s, t ∈ [0, 1], we have (1) x#0 y = x, x#1 y = y, x#t x = x; (2) (x#r y)#t (x#s y) = x#(1−t)r+ts y; (3) x#t (x#s y) = x#ts y. Proof. We use that t → x#t y = expx (tv) = πΦF (t, v), 0 ≤ t ≤ 1 is the unique geodesic from x to y in U0 for (x, y) ∈ DW , y = expx (v). (1) Immediate. (2) For fixed r and s, α(t) = x#(1−t)r+ts y = expx ((1 − t)r + ts)v is continuous, has image contained in W , and satisfies α(0) = x#r y and α(1) = x#s y. Furthermore, for any t, u ∈ [0, 1] α
t+u 2
= expx
= expx
t+u 1−t+1−u r+ s v 2 2 (1 − t)r + ts + (1 − u)r + us 2
v
= expx ((1 − t)r + ts)v # expx ((1 − u)r + us)v = α(t)#α(u),
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where we have used the fact that t → expx (tv) is midpoint preserving (Lemma 3.6). By Proposition 3.8 α is a geodesic, hence must be the unique geodesic in W connecting α(0) = x#r y and α(1) = x#s y. It follows from uniqueness of geodesics in W that α(t) = expx#r y (tw), where expx#r y (w) = x#s y, and hence that (x#r y)#t (x#s y) = α(t) = x#(1−t)r+ts y. (3) Using (1) and (2), we have x#t (x#s y) = (x#0 y)#t (x#s y) = x#(1−t)0+ts y = x#ts y.
2
The Lie–Trotter product formula eA+B = lim
n→∞
A
B
n
en en
for square matrices A and B is of great utility and generalizes to various settings such as semigroups of operators and Lie theory. We now derive the Lie–Trotter formula for the local midpoint operation # on a smooth manifold X with spray. In the following theorem and proof, we fix some distinguished (but arbitrary) point x ∈ X, denote expx : Tx X → X simply by exp and its local inverse (defined on a neighborhood of x) by log (Theorem 2.10), and via a local trivialization identify Tx X with E. Theorem 3.10. For v, w ∈ Tx X, 1 log(exp(2tv)# exp(2tw)) t 2w 2v = lim n log exp # exp . n→∞ n n
v + w = lim
t→0
Proof. Let (W, V ) be a normal neighborhood pair of x. Since (x, x) ∈ DW , there is an open set U containing x such that U × U ⊆ DW . Then the map f (v, w) = log(exp(2v)# exp(2w)), is defined and smooth on a small neighborhood B() of 0 ∈ E. By the definition of the directional derivative for f , we have 1 log(exp(2tv)# exp(2tw)). t→0 t
df(0,0) (v, w) = lim
We compute f (v, 0). The map β(t) = exp(2tv) is a geodesic in W from x to exp(2v), and so x# exp(2v) = exp(v). Thus f (v, 0) = log(exp(2v)#x) = log(x# exp(2v)) = v, and similarly f (0, w) = w. Thus the partial derivatives of f are projections into the respective coordinates. Since the directional derivative is the sum of the partial derivatives, we have
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df(0,0) (v, w) = v + w. Replacing t by
1 , we obtain the second equality. 2 n
4. Locally midpoint-preserving maps In the preceding section we saw that we can define a midpoint operation in a manifold with spray by evaluating a unique geodesic between two points at t = 1/2. In this section we show that continuous maps locally preserving the midpoint operation # in manifolds equipped with a spray are smooth. We first introduce locally midpoint-preserving maps and develop some fundamental properties associated with them. Definition 4.1. For two manifolds X and Y with sprays, we say that a continuous map f : X → Y is a locally midpoint-preserving map if for each x0 ∈ X there exists an open set U containing x0 such that f (x#y) = f (x)#f (y) for all x, y ∈ U . Lemma 4.2. Let T be a subset of [0, 1] containing {0, 1} and closed under the operation of taking midpoint. Then T contains all dyadic rational numbers and if it is closed, then it is equal to [0, 1]. Proof. By hypothesis clearly T contains all dyadic rational numbers between 0 and 1 with denominator 20 and contains all those with denominator 2n+1 if it contains those with denominator 2n . By induction T contains all dyadic rational numbers, and moreover, is dense. If it is closed, therefore, it is all of [0, 1]. 2 Lemma 4.3. Let γ : [0, b] → E, b > 0 be a continuous and midpoint-preserving map with γ(0) = 0 into a topological vector space E. Then there exists v ∈ E such that γ(t) = tv is a linear extension of γ to all of R. Proof. Suppose b = 1. Let v = γ(1) ∈ E. Set the set T = {t ∈ [0, 1] : γ(t) = tv}. Then clearly 0, 1 ∈ T . If s, t ∈ T , then γ
s+t 2
=
1 1 (γ(s) + γ(t)) = (sv + tv) = 2 2
s+t 2
v.
Thus T is closed under taking midpoints. By Lemma 4.2 T = [0, 1] since T is easily seen to be closed. If b = 1 then we first scale by b from [0, 1] to [0, b] and apply the preceding paragraph to the composition of γ with the scaling. 2 Theorem 4.4. Let X, Y be manifolds with sprays and let f : X → Y be a continuous and locally midpointpreserving map. Then f is smooth. Furthermore, if the exponential maps for X, Y are analytic, then f is analytic. Proof. We choose a distinguished point a ∈ X, take f (a) for the distinguished point in Y , and define g locally near 0a ∈ Ta X by g = logf (a) ◦f ◦ expa .
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f
X
Y
expf (a)
expa
Ta X
139
g
Tf (a) Y
Here “locally” means choosing a normal ball B = expa (B(0a , r)) such that f (B) is contained in a normal ball around f (a) and so that log can be defined as the inverse of expf (a) . In this proof we denote {tu : 0 ≤ t ≤ 1} = [0, 1]u and {expa (tu) : 0 ≤ t ≤ 1} = expa ([0, 1]u) for convenience. We proceed in steps. Step 1. We show for u ∈ B(0a , r) there exists w ∈ Tf (a) Y such that g(tu) = tw for all t ∈ [0, 1]. From Lemma 3.6 we have seen the mapping t → expa (tu) is midpoint-preserving, and thus so is tu → expa (tu) from [0, 1]u to expa ([0, 1]u). Let x = expa (u), let y = f (x), and let v = logf (a) (y). Then expf (a) ([0, 1]v) is also closed under the midpoint operation #, and logf (a) is midpoint-preserving from expf (a) ([0, 1]v) to [0, 1]v. The map t → tu → expa (tu) → f (expa (tu)) from [0, 1] into Y is a composition of midpoint-preserving maps, and hence is midpoint-preserving. Since it carries 0a to f (a) and 1 to f (x) = y = expf (a) (v), one argues directly from Lemma 4.2 that all dyadic rational numbers are carried into expf (a) ([0, 1]v), and hence all of [0, 1] are since expf (a) ([0, 1]v) is compact and the composition is continuous. It follows that f (expa ([0, 1]u)) ⊆ expf (a) ([0, 1]v). Combining all this information together, we conclude that g = logf (a) ◦f ◦ expa restricted to [0, 1]u is midpoint-preserving. Thus t → g(tu) is a continuous and midpoint-preserving map carrying 0 to 0f (a) . By Lemma 4.3 there exists w ∈ Tf (a) Y such that g(tu) = tw for all t ∈ [0, 1]. Step 2. We show that we can uniquely u extend g so that g is globally defined, continuous, and positively for any real number m > 0. Let u ∈ Ta X and choose 0 < m < n homogeneous, that is, g(u) = mg m m u u u u < 1. By Step 1, there exists w ∈ Tf (a) Y such such that , ∈ B(0a , r). Then = s , where s = n m n um n that g t = tw for all t ∈ [0, 1]. It follows that m u m su m u ng = g = sw = mw = mg . n s m s m u Thus the definition of g(u) = mg is independent of the real number m > 0, as long as m is sufficiently m u large so that < r. m u For u ∈ Ta X, fix m > 0 such that < r. Then there is an open neighborhood U of u such that m v v , which is continuous. < r for v ∈ U . Then on U the extension of g is given by v → mg m m su For u ∈ Ta X and s > 0, pick m > s such that < r. Then m s su =m g(u) = sg(u). g(su) = mg m m Thus g is positively homogeneous.
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Step 3. We use the Lie–Trotter formula to show that g is additive. We temporarily abbreviate the exponential functions expa and expf (a) by exp, distinguishing them by context, and the corresponding log functions by log. We first use the positive homogeneity and the local equality g = log ◦f ◦ exp in various equivalent forms to calculate for u, v ∈ Ta X and n large: 2u 2v g n log exp # exp n n 2u 2v = n(g ◦ log) exp # exp n n 2v 2u # exp = n(log ◦f ) exp n n 2u 2v = n log f exp #f exp n n 2u 2v = n log exp g # exp g n n 2 2 g(u) # exp g(v) . = n log exp n n We thus have by the Lie–Trotter formula: g(u + v) = g
lim n log exp
n→∞
2u n
# exp
2v n
2u 2v = lim g n log exp # exp n→∞ n n 2 2 g(u) # exp g(v) = lim n log exp n→∞ n n = g(u) + g(v). Step 4. Since g is continuous, additive, and positively homogeneous, it follows easily that it is a continuous linear mapping, hence smooth. Near a, f = expf (a) ◦g ◦ loga and is thus smooth. Since a was an arbitrary choice for the distinguished point, the map f is smooth in a neighborhood of every point, hence smooth. If each exponential map expx is analytic, then the local inverse log is also analytic since the derivative at 0x is invertible (indeed, it is identity). Thus, the composition f = expf (x) ◦g ◦ logx is locally analytic, hence analytic. 2 5. Midpoints in symmetric spaces In this section we consider symmetric spaces in the sense of Loos, review some pertinent results, and then draw connections with the previous material on midpoints. Definition 5.1. We say (X, •) is a Loos symmetric space if X is a smooth Banach manifold, and (x, y) → x • y : X × X → X is a smooth map satisfying for all a, b, c ∈ X: (S1) (S2) (S3) (S4)
a • a = a; a • (a • b) = b; a • (b • c) = (a • b) • (a • c); Every a ∈ X has a neighborhood U such that a • x = x implies a = x for all x ∈ U .
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We view the left translation Sx (y) = x • y as a point reflection or symmetry through x. Properties (S1) and (S2) have obvious geometric interpretation. Remark 5.2. Geometric intuition suggests that reflection through the midpoint m of a and b should carry a to b. In other words, the solution of Sx a = x • a = b should be x = m, provided the midpoint m exists and is unique. Loos symmetric spaces on Banach manifolds have been recently studied by K.-H. Neeb [6], and Lawson and Lim [4]. Let us review some of their results. Theorem 5.3. (See [6].) Let (X, •) be a Loos symmetric space. (i) Identifying T (X × X) with T (X) × T (X), then v • w := T (μ)(v, w) where μ(x, y) := x • y defines a Loos symmetric space on T (X). In each tangent space Tx X, v • w = 2v − w. (ii) The function F : T (X) → T (TX ),
(iii) (iv) (v) (vi)
F (v) := −T (Sv/2 ◦ Z)(v)
defines a spray on X, where Z : X → TX is the zero section and Sv/2 is the point symmetry for v/2 from part (i). Aut(X, •) = Aut(X, F ), where the former consists of all diffeomorphisms that are automorphisms with respect to • and the latter consists of all diffeomorphisms that preserve the spray F . F is uniquely defined as the only spray invariant under all symmetries Sx , x ∈ X. Every geodesic of (X, F ) extends to a geodesic defined on all of R. Let α : R → X be a geodesic and call the maps τα,s := Sα(s/2) ◦ Sα(0) , s ∈ R, translations along α. Then these are automorphisms of (X, •) with τα,s (α(t)) = α(t + s)
and
dτα,s (α(t)) = Ptt+s (α)
for all s, t ∈ R. We write Ptt+s (α) : Tα(t) X → Tα(t+s) X for the linear map given by parallel transport along α. The set of real numbers R endowed with the core operation s • t = s + (−t) + s = 2s − t defines a Loos symmetric space. Proposition 5.4. (See [4].) The (maximal) geodesics of a Loos symmetric space (X, •) equipped with its unique symmetry-invariant spray are precisely the continuous homomorphisms β : (R, •) → (X, •), which are each of the form βv (t) = expx (tv), where x ∈ X, v ∈ Tx X, and hence are smooth. The correspondence v ↔ βv is a one-to-one correspondence between Tx X and all maximal geodesics taking on the value x at 0. We recall some examples of sprays on Loos symmetric spaces.
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Example 5.5. Let E be a Banach space with trivial tangent bundle E ×E. Consider that (E, •) is a symmetric space with x •y = 2x −y. Then the tangent bundle T (E) becomes a Loos symmetric space by Theorem 5.3(1). If we denote a point in the tangent bundle by a pair (x, v) in E × E, then we have
S(x, v2 ) ◦ Z (y) = S(x, v2 ) (y, 0) = (2x − y, v),
and so, T (S(x, v2 ) ◦ Z)(y, w) = (2x − y, v, −w, 0) ∈ (X × E) × (E × E). Therefore, F (x, v) = −T (S(x, v2 ) ◦ Z)(x, v) = (x, v, v, 0). Example 5.6. The open convex cone Ω of all positive definite operators on a Hilbert space is a Loos symmetric space with respect to the core operation μ(A, B) := A • B = AB −1 A. Then the tangent bundle T Ω can be denoted by T Ω = Ω × Herm, where Herm is the space of hermitian operators. Moreover, (T Ω, T μ) becomes a Loos symmetric space, where T μ((A, X), (B, Y )) = (A • B, XB −1 A + AB −1 X − AB −1 Y B −1 A). This implies that
1 S(A, X ) ◦ Z (B) = S(A, X ) (B, 0) = A • B, (XB −1 A + AB −1 X) . 2 2 2
By [6, Lemma 3.2] the first coordinate of the differential d(S(A, X ) ◦ Z)(A) is −idTA (Ω) , and the second 2 coordinate at X is 1 (−XB −1 XB −1 A − AB −1 XB −1 X). 2 So 1 −1 −1 T (S(A, X ) ◦ Z)(A, X) = A • A, X, −X, (−XA X − XA X) 2 2 = (A, X, −X, −XA−1 X). Therefore, the canonical spray is given by F (A, X) = −T (S(A, X ) ◦ Z)(A, X) = (A, X, X, XA−1 X). 2
On a Loos symmetric space, we can define the midpoint locally (as in Section 3) by means of the exponential map associated with the canonical spray. In the next proposition we show that this midpoint can be alternatively characterized as a midpoint of symmetry, i.e., a point m = m(x, y) that satisfies Sm x = m • x = y and Sm y = m • y = x.
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Proposition 5.7. Let (X, •) be a Loos symmetric space. Then the midpoint locally defined in DW can be alternatively characterized as the unique midpoint of symmetry in W . Proof. Let x0 ∈ X and let (W, V ) be a normal neighborhood pair for x0 (Theorem 2.12). For any pair (x, y) ∈ DW , we have a geodesic βv (t) = expx (tv) with βv (0) = x, βv (1) = y and βv ([0,1]) ⊆ W . According 1 v . Furthermore, to Definition 3.2, the midpoint m of x and y is given by m = x#y = βv (1/2) = expx 2 by Proposition 5.4 y = βv (1) = βv (2(1/2) − 0) = βv
1 •0 2
1 = βv • βv (0) = m • x. 2
Suppose m ∈ W also satisfies y = m • x. By Theorem 2.12(2)(ii), the exponential expm maps the open subset {m } × B() of Tm X diffeomorphically onto an open set U (m ) containing W and contained in U0 . Let expm (−w) = x and set βw (t) = expm (tw). By Proposition 5.4, βw is defined on all of R and is a homomorphism with respect to the operation •. Thus βw (1) = βw (0 • −1) = βw (0) • βw (−1) = m • x = y. Setting γ(t) = βw (2t −1), we obtain a geodesic on [0, 1] running from x to y (Remark 2.8), and by uniqueness of geodesics in U0 , we obtain γ(t) = x#t y for 0 ≤ t ≤ 1, in particular, x#y = γ(1/2) = βw (0) = m . 2 It follows immediately from Proposition 5.7 that in a smooth Loos symmetric space the operations • and # locally determine each other in the sense that locally m •x = y if and only if m = x#y. Thus a continuous map between two such symmetric spaces locally preserves one of the operations (see Definition 4.1) if and only if it locally preserves the other. We thus have the following corollary to Theorem 4.4. Corollary 5.8. Let (X, •) and (Y, •) be smooth Loos symmetric spaces, and let h : X → Y be a continuous map that is locally a homomorphism with respect to • at each point of X. Then h is smooth. The conclusion holds in particular if h is a continuous •-homomorphism. 6. Midpoints in Riemannian manifolds In this section we review how to construct unique local midpoints from the distance function associated with a smooth Riemannian manifold modeled on a finite or infinite dimensional Hilbert space. Let (X, g) be a connected Riemannian manifold, a smooth connected manifold X with a smooth section g of positive-definite quadratic forms on the tangent bundle giving each tangent space the structure of a Hilbert space. We recall some fundamental results about Riemannian manifolds, see Chapters VII–IX in [2]. For any piecewise smooth curve α : [a, b] → X, an arc length L(α) is given by b L(α) =
α (t) dt,
a
where α (t) = α (t), α (t)α(t) . The g-distance or simply distance δ is defined by setting δ(x, y) equal to the arc length infimum over all piecewise smooth curves from x to y. The distance function δ is indeed a metric, and the corresponding metric topology agrees with the original topology on X (see Proposition 6.1 in [2, Chapter VII]). Any Riemannian manifold X has a unique connection associated with the metric g, called the Levi-Civita connection or the canonical connection. This is typically used to define the geodesics of the Riemannian 1/2
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manifold, although in [2, Section VII.7] S. Lang has defined them alternatively (and equivalently) from the canonical spray associated to a Riemannian structure. The map t → exp(tv) for tv ∈ D, where D is the domain of the exponential map as in Section 2, gives the unique maximal geodesic that passes through x = exp(0x ) at t = 0 with velocity v ∈ Tx X. For v ∈ Tx X the geodesic t → exp(tv) has constant speed v , and hence for v ∈ D, this geodesic restricted to [0, 1] has length v . A geodesic β : [a, b] → X is called a minimal geodesic if L(β) ≤ L(α) for every piecewise smooth curve α joining β(a) = x to β(b) = y. This means equivalently that L(β) = δ(x, y). Since the geodesic β has some constant speed σ, we have δ(x, y) = L(β) = σ(b − a). It is straightforward to see that the restriction of β to any closed subinterval of [a, b] will again be a minimal geodesic. An open set U of X is said to be convex if given x, y ∈ U there exists a unique minimal geodesic in U joining x and y, where uniqueness means unique up to reparameterization. The following is a form of Whitehead’s theorem (see Theorem 5.8 and Corollary 5.3 in [2, Chapter VII]). Theorem 6.1. Let (X, g) be a Riemannian manifold, and let p ∈ X, Then there exists > 0 such that the following are satisfied: (1) For 0 < r ≤ , the open neighborhood B(p, r) = expp Bg (0p , r) is convex, where the second ball is taken in Tp X in the norm arising from g. (2) The map expp restricted to Bg (0p , ) is a diffeomorphism. (3) For 0 < r ≤ , the mapping (x, y) → v, where v ∈ Tx X and t → expx (tv) for 0 ≤ t ≤ 1 is the minimal geodesic joining x and y, is a diffeomorphism from B(p, r) × B(p, r) onto an open subset of D, the domain of exp. (4) For any v ∈ Bg (0p , ), the mapping tv → exp(tv) for 0 ≤ t ≤ 1 is an isometry. (5) For 0 < r ≤ /2 and x, y ∈ B(p, r), there is exactly one metric midpoint (with respect to the distance δ) between x and y. For metric spaces with distance δ, a point m is said to be a (metric) midpoint of x and y if δ(x, m) = δ(m, y) =
1 δ(x, y). 2
In general points x and y may have no, one, or several midpoints. However, there is only one in the case that there exists a unique minimal geodesic connecting x to y. Indeed, let β be a parameterization of this minimal geodesic with domain [a, b] and constant speed σ, where β(a) = x and β(b) = y. Then for c = (a + b)/2 and m = β(c), δ(x, m) = L(β|[a,c] ) = σ(c − a) =
1 1 σ(b − a) = δ(x, y), 2 2
where m = β(c). Similarly δ(m, y) = (1/2)δ(x, y), and no other point on the geodesic is a midpoint. We denote this uniquely determined midpoint for pairs (x, y) that lie on a unique minimal geodesic by x#y. In the next proposition we show that this midpoint can be characterized as a midpoint locally defined by the exponential map associated with the canonical spray. Proposition 6.2. Let (X, g) be a connected Riemannian manifold. Then the geodesic midpoint locally defined in DW (from the associated spray or Levi-Civita connection) can be characterized as the unique local metric midpoint in terms of Riemannian distance δ.
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Proof. For any pair (x, y) ∈ DW let β(t) = expx (tv), 0 ≤ t ≤ 1, be a geodesic with β(0) = x and β(1) = y. By Theorem 6.1(3) the map β is the unique minimal geodesic joining x and y. Hence by our previous discussion x#y is the image of geodesic β at t = 1/2, which means it agrees with the midpoint locally defined by the exponential map associated with the canonical spray. 2 Combining together Proposition 6.2 and Theorem 4.4, we obtain the following corollary; see Theorem 4.3 of [3]. Corollary 6.3. Let X, Y be smooth Riemannian manifolds and let σ : X → Y be a continuous and locally midpoint preserving map. Then σ is smooth. Furthermore, if the exponential maps for X, Y are analytic, then σ is analytic. We close by returning to a special case of Example 5.6. Example 6.4. We consider the manifold M of n × n-positive definite Hermitian matrices. There is a natural and well-known Riemannian metric, or called the Riemannian trace metric, on M making it a Riemannian manifold, indeed a symmetric space, of negative curvature (see, e.g., [5]). The corresponding distance trace n 1 metric between two positive definite matrices is given by δ(A, B) = ( i=1 log2 λi (A−1 B)) 2 , where λi (X) denotes the ith eigenvalue of X in non-decreasing order. By [5] the metric midpoint (with respect to the distance trace metric) agrees with the symmetric midpoint, which is the solution X of XA−1 X = X • A = B (see Example 5.6). By elementary algebra, one sees that the unique positive definite solution is given by A#B = A1/2 (A−1/2 BA−1/2 )1/2 A1/2 , the geometric mean of the two positive definite matrices A, B. The local validity of Proposition 5.7 extends to the global case in this setting (two distinct points uniquely determine a globally defined geodesic), so that the midpoints are also the geodesic midpoints for the canonical spray associated with the symmetric space structure. Hence our three methods for defining midpoints (geodesic, symmetric, and metric) all converge. 7. Closing comments The material presented in this paper applies also to certain settings in Finsler geometry, although we have not as of yet systemically worked out the details nor the range of applications. What one needs are a Banach manifold M equipped with both a spray and a Finsler structure that are appropriately interrated in the following sense: • parallel transport along the geodesics of the spray are isometries of the tangent spaces equipped with their Finsler norms; • each point of M has a neighborhood for which the geodesics of the spray are minimal geodesics for the Finsler metric, i.e., the Finsler distance between two points x, y in the neighborhood is the Finsler length of the unique geodesic of the spray contained in the neighborhood (see Theorem 2.12). (We remark however that even locally the minimal geodesics for the Finsler spray need not be unique – we are only requiring that the spray geodesic be one of them.) One can generalize much the preceding material on Riemannian manifolds to this setting. One shows that in a neighborhood Nx of 0x , x ∈ M , the map t → expx (tv), v ∈ Nx , from [0, 1] into Nx is an isometry and so (1/2)v maps to the metric midpoint of the image. It follows that locally midpoints with respect to the spray are also midpoints with respect to the Finsler metric, a weakened version of Proposition 6.2.
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Specific settings where the previous conditions are satisfied include Finsler symmetric spaces of nonpositive curvature modeled on general Banach spaces including the open cone of positive invertible elements for a C ∗ -algebra [6], [4, Sections 8,9] and the Hilbert metric on an open bounded convex subset of Rn [8]. References [1] [2] [3] [4] [5] [6] [7] [8]
R. Howe, Very basic Lie theory, Am. Math. Mon. 90 (1983) 600–623. S. Lang, Fundamentals of Differential Geometry, Springer, 1999. J. Lawson, Y. Lim, A Lie–Trotter formula for Riemannian manifolds and applications, J. Lie Theory 20 (2010) 665–672. J. Lawson, Y. Lim, Symmetric spaces with convex metrics, Forum Math. 19 (2007) 571–602. J.D. Lawson, Y. Lim, The geometric mean, matrices, metrics, and more, Am. Math. Mon. 108 (2001) 797–812. K. Neeb, A Cartan–Hadamard theorem for Banach–Finsler manifolds, Geom. Dedic. 95 (2002) 115–156. H.F. Trotter, On the product of semi-groups of operators, Proc. Am. Math. Soc. 10 (1959) 545–551. C. Vernicos, Introduction aux géométries de Hilbert, Sémin. Théor. Spectr. Géom. 25 (2005) 145–168.
Contents lists available at ScienceDirect
Differential Geometry and its Applications www.elsevier.com/locate/difgeo
Local midpoints on smooth manifolds Sejong Kim a , Jimmie Lawson b,∗ a b
Department of Mathematics, Chungbuk National University, Cheongju, 361-763, Republic of Korea Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
a r t i c l e
i n f o
Article history: Received 6 March 2014 Available online xxxx Communicated by B. Ørsted MSC: 53B15 53C35 53B20
a b s t r a c t In this paper we consider three methods for obtaining midpoints, primarily midpoints of geodesics of sprays, but also midpoints of symmetry (in symmetric spaces), and metric midpoints (in Riemannian manifolds). We derive general conditions under which these approaches yield the same result. We also derive a version of the Lie–Trotter formula based on the midpoint operation and use it to show that continuous maps preserving (local) midpoints are smooth. © 2015 Elsevier B.V. All rights reserved.
Keywords: Midpoint Spray Lie–Trotter formula Smooth manifold Symmetric space Riemannian manifold
1. Introduction There are various natural notions of a “midpoint” that present themselves. The first is a geodesic approach: given a notion of a geodesic α that is uniquely minimal between a = α(t0 ) and b = α(t1 ), one may take as midpoint α((t0 + t1 )/2). The second is a metric approach: given a, b in a metric space, we may take as midpoint a point m such that d(a, m) = d(m, b) = (1/2)d(a, b), provided such an m uniquely exists. The third approach is what we might call a midpoint of symmetry: given a set X endowed with a geometric structure that includes a point reflection Sa : X → X through each point a, we may define a midpoint m of a and b to be one satisfying Sm (a) = b and vice-versa, again provided such an m uniquely exists. The principal approach of this paper is to consider smooth manifolds equipped with sprays, a variant of connections. The sprays give rise to geodesics, which in small enough neighborhoods are unique. We review and slightly rework this machinery in Section 2, since it is crucial to what follows. In Section 3 we define * Corresponding author. E-mail addresses: [email protected] (S. Kim), [email protected] (J. Lawson). http://dx.doi.org/10.1016/j.difgeo.2015.01.009 0926-2245/© 2015 Elsevier B.V. All rights reserved.
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midpoints locally via the geodesic approach as in the preceding paragraph. Indeed we can define weighted means a#t b for 0 ≤ t ≤ 1, with the midpoint corresponding to t = 1/2. We further show in Section 3 that the local midpoint operation satisfies a variant Lie–Trotter product formula based on the local binary midpoint operation. The standard Lie–Trotter product formula n eA+B = lim eA/n eB/n n→∞
first formulated by Sophus Lie for square matrices A and B is of great utility and generalizes to various settings such as semigroups of (unbounded) operators [7] and Lie theory. For an elementary derivation and basic applications of the formula in the setting of Lie theory, we refer the reader to [1]. The famous fifth problem of Hilbert asked whether a locally euclidean topological group must have a differentiable multiplication, i.e., must be a Lie group. More generally the problem asked for general conditions for a continuous function to be smooth. As an application of our results in the fourth section we apply our Lie–Trotter formula to show this is true (in the category of smooth manifolds equipped with sprays) for continuous functions preserving the (local) midpoint operation. In Section 5 we consider midpoints of symmetry in the context of smooth symmetric spaces modeled on general Banach spaces. Using a crucial result of K.-H. Neeb [6] that a smooth symmetric space uniquely determines a canonical spray, we establish that locally the midpoints of symmetry arising from the symmetric space structure agree with the geodesic midpoints of the associated spray. In Section 6 we turn to Riemannian manifolds. Any Riemannian manifold X has a unique connection associated with the Riemannian metric g, called the Levi-Civita connection or the canonical connection. This is typically used to define the geodesics of the Riemannian manifold, although in [2, Section VII.7] Lang has defined them alternatively (and equivalently) from the canonical spray associated to a Riemannian structure. We show that locally the metric midpoints with respect to the distance function determined by g agree with the geodesic midpoints of the associated canonical spray associated with the Levi-Civita connection. We close this section with the illustrative example of the symmetric space of all positive definite matrices, where our three approaches to defining midpoints all converge. In the final section we indicate briefly how our results generalize to the setting of Finsler manifolds. 2. Sprays and exponential maps We work in the category of smooth manifolds modeled on Banach spaces and smooth maps, where smooth means C ∞ . The tangent bundle of a smooth manifold X is denoted π : TX → X. A vector field on X means a smooth map ξ : X → TX such that ξ(x) lies in the tangent space Tx X for each x ∈ X, i.e., π ◦ ξ = idX . An integral curve for ξ with initial condition x0 ∈ X is a smooth curve α : J → X mapping an open interval J ⊆ R containing 0 into X such that for all t ∈ J α (t) = ξ(α(t)), and α(0) = x0 .
(2.1)
From Theorem 2.2 in [2, Chapter IV.2], we obtain the following result. Lemma 2.1. Let ξ : X → TX be a smooth vector field. Then there exists a smooth map Φ : U → X, where U is an open subset of R × X such that {0} × X ⊆ U , satisfying the following. (1) Φ(0, x) = x.
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(2) Φ(s, Φ(t, x)) = Φ(s + t, x) provided Φ(t, x) is defined and either the left-hand or right-hand side is defined. (3) Let J(x) = (t− (x), t+ (x)) be the largest time interval for x containing 0 such that (t− (x), t+ (x)) ×{x} ⊆ U = dom(Φ). Then ϕx (t) on J(x) defined by ϕx (t) = Φ(t, x) is a maximal integral curve, in particular ϕx (t) = ξ(ϕx (t)). Furthermore, the solution ϕx of (2.1) is unique on any interval on which it is defined with ϕx (0) = x. The smooth map Φ in Lemma 2.1 is called a local flow, item (1) the identity property, and item (2) the semigroup property. Definition 2.2. A second-order vector field is a vector field F : TX → T (TX ) satisfying T (π) ◦ F = idTX . Applying Lemma 2.1 to a second-order vector field F , we obtain a local flow ΦF : UF → TX , where UF is an open subset of R × TX containing {0} × TX , such that for any vector v ∈ TX (1) ΦF (0, v) = v, (2) ΦF (s, ΦF (t, v)) = ΦF (s + t, v) provided ΦF (t, v) is defined and either the left-hand or right-hand side is defined. The condition for a vector field to be a second-order vector field can be stated alternatively in terms of the flow of the vector field, see [2, Chapter IV.3]. Proposition 2.3. Let F : TX → T (TX ) be a vector field. Then F is a second-order vector field if and only if the local flow ΦF on the tangent space TX satisfies (π ◦ ΦF ) = ΦF , more specifically, ∂ (π ◦ ΦF )(t, x) = ΦF (t, x). ∂t Definition 2.4. Let s ∈ R and s∗ : TX → TX denote the scalar multiplication by s in each tangent space. A second-order vector field F on TX is called a spray if for all s ∈ R and v ∈ TX F (sv) = T (s∗ )(sF (v)).
(2.2)
In [2, Chapter IV.4] S. Lang presents alternative equivalent conditions for defining a spray in terms of the integral curves of the second-order vector field F . We translate these to equivalent conditions on the local flow. Proposition 2.5. Let ΦF be the local flow of the second-order vector field F with initial condition v. Then for v ∈ TX , each of the following is equivalent to the condition (2.2): (1) A pair (t, sv) is in the domain of ΦF if and only if (st, v) is in the domain of ΦF , and then ΦF (t, sv) = sΦF (st, v).
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(2) A pair (st, v) is in the domain of ΦF if and only if (s, tv) is in the domain of ΦF , and then πΦF (s, tv) = πΦF (st, v). (3) A pair (t, v) is in the domain of ΦF if and only if (1, tv) is in the domain of ΦF , and then πΦF (t, v) = πΦF (1, tv). Lemma 2.6. Given a spray F : TX → TTX , a C 1 -map α : J → X from some interval J ⊆ R, and v ∈ TX , the following are equivalent: (1) for all t ∈ J, α(t) = πΦF (t, v); (2) α = π ◦ ϕv |J , where ϕv (t) = ΦF (t, v) for t ∈ J(v); (3) for all t ∈ J, α (t) = ΦF (t, v). Under these equivalent hypotheses α is the composition of smooth functions, hence smooth. Proof. One sees directly that (1) and (2) are equivalent. That (1) implies (3) follows from Proposition 2.3. Assume (3) holds. Then for t ∈ J, πΦF (t, v) = π(α (t)) = α(t), so (1) is satisfied. The last assertion follows from (2). 2 Definition 2.7. A geodesic of a spray F is a (necessarily smooth) map α : J → X from some interval J ⊆ R satisfying any of the equivalent conditions of Lemma 2.6. Remark 2.8. For a geodesic α, a reparametrization β(t) = α(bt + c) is again a geodesic since β(t) = α(bt + c) = πΦF (bt + c, v) = πΦF (bt, ΦF (c, v)) = πΦF (t, bΦF (c, v)). We now recall how the local flow ΦF of a spray F gives rise to its associated exponential map. Let D = {v ∈ TX : [0, 1] × {v} ⊆ dom(ΦF )}. We know from Corollary 2.7 in [2, Chapter IV.2] that D is an open set in TX , and the map v → ΦF (1, v) is a smooth map of D into TX . We define the exponential map exp : D → X, exp(v) = πΦF (1, v) and call D the domain of the exponential map associated with the spray F . Note that the map exp is smooth. If x ∈ X and 0x denotes the zero vector in Tx X, then we obtain exp(0x ) = x. Indeed, exp(0x ) = πΦF (1, 0x ) = πΦF (1, 0 · 0x ) = πΦF (0, 0x ) = π(0x ) = x. Thus the exponential map is defined and coincides with π on the zero section, and so induces an diffeomorphism of the cross section onto X. We denote the restriction of exp to the tangent space Tx X by expx : expx : D ∩ Tx X → X.
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Remark 2.9. (1) From the preceding we can define expx (tv) for v in the domain of expx and t ∈ dom(ϕv ), in particular for 0 ≤ t ≤ 1, as expx (tv) = πΦF (1, tv) = πΦF (t, v) = πϕv (t). It follows for v ∈ dom(expx ) that t → expx (tv) is a geodesic on an open interval containing [0, 1]. Furthermore, we have expx (s(tv)) = πΦF (s, tv) = πΦF (st, v) = expx ((st)v) by Proposition 2.5(2) for (st, v) in the domain of ΦF , in particular for 0 ≤ s ≤ 1. (2) For v in the domain of expx , define α(t) = expx (t) and β(t) = α(1 − t) for 0 ≤ t ≤ 1. Then β(t) = expx (1 − t) = πΦF (1 − t, v) = πΦF (−t, ΦF (1, v)) = πΦF (t, −ΦF (1, v)), where the third equality follows from the fact ΦF is a local flow and the fourth from Proposition 2.5(2). It follows from this equation that β is a geodesic. We recall two important facts about the exponential function, first Theorem 4.1 of [2, Section IV.4]. Theorem 2.10. The derivative of the exponential map expx at 0x is the identity, and hence it is a local diffeomorphism from Tx X into X on a neighborhood of 0x . The second fact follows from Proposition 4.2 of [2, Section IV.4]. Proposition 2.11. For any v ∈ Tx X at which expx is defined, the map αv (t) = expx (tv) is a geodesic. Conversely if α : J → X is a geodesic with α(0) = x and α (0) = v ∈ Tx X, then α(t) = expx (tv) for all t ∈ J. For > 0 we denote by B() the open ball of elements v ∈ E, the Banach space on which X is modeled, with v < . Let x0 ∈ X. From the definition of the tangent space and the resulting local triviality of the tangent bundle, a chart defined on a neighborhood U of x0 induces a commuting triangle π −1 (U ) π
U ×E πU
U where the horizontal arrow is a diffeomorphism and a vector space isomorphism on the fibers and πU is projection into the first coordinate. It follows that if we work only in the neighborhood U , we can take the tangent bundle TU to be U × E with bundle map first projection. From the results of [2, Chapter VIII.5], we can deduce for an arbitrary point x0 ∈ X the existence of a well-behaved neighborhood W in the sense of the following theorem. Theorem 2.12. (1) Given x0 ∈ X, there exists an open set V ⊆ TX containing 0x0 such that G(v) = (πv, exp v) is a diffeomorphism from V onto some open subset of X × X containing (x0 , x0 ). We may pick V such that for some > 0 in some local trivialization of the tangent bundle V = U0 × B() ⊆ U0 × E = TU 0 .
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(2) There further exists an open neighborhood W of x0 , W ⊆ U0 , such that G(V ) ⊇ W ×W and the following additional properties are satisfied: (i) Any two points x, y ∈ W are joined by a unique geodesic t → expx (tv), 0 ≤ t ≤ 1 lying in U0 , where expx (v) = y. This geodesic depends smoothly on the pair (x, y) in the sense that the correspondence v ↔ G(v) = (πv, exp v) = (x, expx v) = (x, y) is a C ∞ -diffeomorphism between an open subset of V and W × W . (ii) For each x ∈ W the exponential expx maps the open subset {x} × B() of Tx X diffeomorphically onto an open set U (x) containing W and contained in U0 . We call the pair (W, V ) a normal neighborhood pair of x0 in X and B() a normal ball around 0x0 in Tx0 X. Remark 2.13. The geodesic α(t) = expx (tv), 0 ≤ t ≤ 1, of Theorem 2.12 is unique in the sense that it is the only geodesic map from [0, 1] into U0 such that α(0) = x and α(1) = y, where y = expx (v). 3. Means and the Lie–Trotter formula Let X be a smooth manifold with spray F . We define locally the weighted means of any pair (x, y), call the equally weighted mean the midpoint, and derive the Lie–Trotter formula with respect to this local midpoint operation. Let x0 ∈ X and let (W, V ) be a normal neighborhood pair of x0 , where V = U0 ×B(), as in Theorem 2.12. From Theorem 2.12 any two points x, y ∈ W are joined by a unique geodesic in U0 , namely t → expx (tv) with x = expx (0x ) = expx (0 · v) and y = expx (v). Unfortunately, for some s ∈ [0, 1] the image expx (sv) may lay outside the normal neighborhood W . So we consider the set DW = {(x, y) ∈ W × W : expx (tv) ∈ W for all 0 ≤ t ≤ 1, where expx (v) = y}. Lemma 3.1. The set DW is open. Proof. For any pair (x, y) ∈ DW , we define a map α : [0, 1] × W × W → U0 by α(t, x, y) = expx (tv), where 0 ≤ t ≤ 1. From the definition of DW we have that α ([0, 1] × {x} × {y}) ⊆ W . Furthermore, the map α is continuous on [0, 1] × W × W by Theorem 2.12 and the continuity of the flow Φ. By the compactness of the factors of [0, 1] × {x} × {y}, there are open sets O1 in W containing {x} and O2 in W containing {y} such that α ([0, 1] × O1 × O2 ) ⊆ W . This means that O1 ×O2 is an open set in DW containing {x} ×{y}. 2 For any pair (x, y) ∈ DW we can naturally define the weighted mean of x and y. Definition 3.2. The t-weighted mean of a pair (x, y) ∈ DW is given by x#t y = expx (tv),
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where 0 ≤ t ≤ 1, expx (v) = y, and t → expx (tv) is the unique geodesic in W from x to y. We write x#y for x# 12 y and call it the midpoint of x and y. Remark 3.3. We note that in a neighborhood of (t, x, y) for (x, y) ∈ DW , we have x#t y = expx (tG−1 (x, y)) = πΦF (1, tG−1 (x, y)), which is a composition of smooth functions (see Theorem 2.12(2)(i)) where defined, hence smooth. Remark 3.4. From Definition 3.2 we see that x# expx (tv) = expx ((t/2)v) for any t ∈ [0, 1]. Indeed, for u = tv, expx (u) ∈ W ; moreover, (x, expx (u)) ∈ DW since expx (su) = expx (stv) ∈ W for 0 ≤ s ≤ 1. Then x# expx (u) = expx
1 u 2
= expx
t v . 2
Similarly we can define the midpoint y#x for a pair (y, x) ∈ DW by y#x := expy
1 w , 2
where x = expy (w). Proposition 3.5. For any pair (x, y) ∈ DW and 0 ≤ t ≤ 1, x#t y = y#1−t x. In particular, x#y = y#x. Proof. Let α(t) = expx (tv), where y = expx (v), be a geodesic from x to y lying entirely in W . By Remark 2.9(2) β(t) = α(1−t) is a geodesic from y to x, which by definition also lies in W . By Theorem 2.12(2)(i) β(t) = expy (tu) for some u for 0 ≤ t ≤ 1. Thus y#1−t x = expy ((1 − t)u) = β(1 − t) = α(t) = expx (tv) = x#t y.
2
In the next sequence of results we show that the t-weighted mean map α(t) = expx (tv) is uniquely determined by its midpoint preservation property. Lemma 3.6. For any pair (x, y) ∈ DW with y = expx (v), the map α(t) = expx (tv), 0 ≤ t ≤ 1, is midpointpreserving from [0, 1] into W . Proof. Let 0 ≤ t1 < t2 ≤ 1, and let s = t2 − t1 . Set w = sΦF (t1 , v), x1 = π(w), and x2 = expx1 (w). We note that x1 = π(w) = π(sΦF (t1 , v)) = π(ΦF (t1 , v)) = expx (t1 v) and x2 = expx1 (w) = πΦF (1, w) = πΦF (1, sΦF (t1 , v)) = πΦF (s + t1 , v) = expx (t2 v). The following calculation now establishes midpoint preservation:
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expx
t 1 + t2 2
s s v = expx t1 + = πΦF t1 + , v 2 2 = πΦF (s/2, ΦF (t1 , v)) = πΦF (1/2, sΦF (t1 , v)) 1 w = πΦF (1/2, w) = expx1 2 2
= x1 #x2 = expx (t1 v)# expx (t2 v). The following corollary is a restatement of the previous lemma.
Corollary 3.7. For (x, y) ∈ DW , s, t ∈ [0, 1], (x#t y)#(x#s y) = x#r y, where r = (s + t)/2. Proposition 3.8. For any pair (x, y) ∈ DW , the map α(t) = expx (tv), 0 ≤ t ≤ 1, with expx (v) = y is the unique midpoint-preserving continuous map from [0, 1] into W with α(0) = x and α(1) = y. Proof. Let β : [0, 1] → W be continuous, midpoint-preserving, and satisfy β(0) = x, β(1) = y. Then by midpoint preservation β(1/2) = β
0+1 2
= β(0)#β(1) = x#y.
Thus (β(0), β(1/2)) = (x, x#y), (β(1/2), β(1)) = (x#y, y) ∈ DW . From Corollary 3.7 1 1 0+ = β(0)#β(1/2) = (x#0 y)#(x#y) = x# 14 y. β(1/4) = β 2 2 By a similar argument on the interval [1/2, 1], we conclude β(3/4) = x# 34 y. Inductively bisecting intervals and applying the same argument, we conclude that β(r) = x#r y for all dyadic rationals r ∈ [0, 1]. Thus β agrees with expx (rv) = x#r y for all dyadic rationals r and by continuity for all 0 ≤ t ≤ 1. 2 We establish some useful identities involving the weighted mean. Proposition 3.9. For any pair (x, y) ∈ DW , and r, s, t ∈ [0, 1], we have (1) x#0 y = x, x#1 y = y, x#t x = x; (2) (x#r y)#t (x#s y) = x#(1−t)r+ts y; (3) x#t (x#s y) = x#ts y. Proof. We use that t → x#t y = expx (tv) = πΦF (t, v), 0 ≤ t ≤ 1 is the unique geodesic from x to y in U0 for (x, y) ∈ DW , y = expx (v). (1) Immediate. (2) For fixed r and s, α(t) = x#(1−t)r+ts y = expx ((1 − t)r + ts)v is continuous, has image contained in W , and satisfies α(0) = x#r y and α(1) = x#s y. Furthermore, for any t, u ∈ [0, 1] α
t+u 2
= expx
= expx
t+u 1−t+1−u r+ s v 2 2 (1 − t)r + ts + (1 − u)r + us 2
v
= expx ((1 − t)r + ts)v # expx ((1 − u)r + us)v = α(t)#α(u),
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where we have used the fact that t → expx (tv) is midpoint preserving (Lemma 3.6). By Proposition 3.8 α is a geodesic, hence must be the unique geodesic in W connecting α(0) = x#r y and α(1) = x#s y. It follows from uniqueness of geodesics in W that α(t) = expx#r y (tw), where expx#r y (w) = x#s y, and hence that (x#r y)#t (x#s y) = α(t) = x#(1−t)r+ts y. (3) Using (1) and (2), we have x#t (x#s y) = (x#0 y)#t (x#s y) = x#(1−t)0+ts y = x#ts y.
2
The Lie–Trotter product formula eA+B = lim
n→∞
A
B
n
en en
for square matrices A and B is of great utility and generalizes to various settings such as semigroups of operators and Lie theory. We now derive the Lie–Trotter formula for the local midpoint operation # on a smooth manifold X with spray. In the following theorem and proof, we fix some distinguished (but arbitrary) point x ∈ X, denote expx : Tx X → X simply by exp and its local inverse (defined on a neighborhood of x) by log (Theorem 2.10), and via a local trivialization identify Tx X with E. Theorem 3.10. For v, w ∈ Tx X, 1 log(exp(2tv)# exp(2tw)) t 2w 2v = lim n log exp # exp . n→∞ n n
v + w = lim
t→0
Proof. Let (W, V ) be a normal neighborhood pair of x. Since (x, x) ∈ DW , there is an open set U containing x such that U × U ⊆ DW . Then the map f (v, w) = log(exp(2v)# exp(2w)), is defined and smooth on a small neighborhood B() of 0 ∈ E. By the definition of the directional derivative for f , we have 1 log(exp(2tv)# exp(2tw)). t→0 t
df(0,0) (v, w) = lim
We compute f (v, 0). The map β(t) = exp(2tv) is a geodesic in W from x to exp(2v), and so x# exp(2v) = exp(v). Thus f (v, 0) = log(exp(2v)#x) = log(x# exp(2v)) = v, and similarly f (0, w) = w. Thus the partial derivatives of f are projections into the respective coordinates. Since the directional derivative is the sum of the partial derivatives, we have
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df(0,0) (v, w) = v + w. Replacing t by
1 , we obtain the second equality. 2 n
4. Locally midpoint-preserving maps In the preceding section we saw that we can define a midpoint operation in a manifold with spray by evaluating a unique geodesic between two points at t = 1/2. In this section we show that continuous maps locally preserving the midpoint operation # in manifolds equipped with a spray are smooth. We first introduce locally midpoint-preserving maps and develop some fundamental properties associated with them. Definition 4.1. For two manifolds X and Y with sprays, we say that a continuous map f : X → Y is a locally midpoint-preserving map if for each x0 ∈ X there exists an open set U containing x0 such that f (x#y) = f (x)#f (y) for all x, y ∈ U . Lemma 4.2. Let T be a subset of [0, 1] containing {0, 1} and closed under the operation of taking midpoint. Then T contains all dyadic rational numbers and if it is closed, then it is equal to [0, 1]. Proof. By hypothesis clearly T contains all dyadic rational numbers between 0 and 1 with denominator 20 and contains all those with denominator 2n+1 if it contains those with denominator 2n . By induction T contains all dyadic rational numbers, and moreover, is dense. If it is closed, therefore, it is all of [0, 1]. 2 Lemma 4.3. Let γ : [0, b] → E, b > 0 be a continuous and midpoint-preserving map with γ(0) = 0 into a topological vector space E. Then there exists v ∈ E such that γ(t) = tv is a linear extension of γ to all of R. Proof. Suppose b = 1. Let v = γ(1) ∈ E. Set the set T = {t ∈ [0, 1] : γ(t) = tv}. Then clearly 0, 1 ∈ T . If s, t ∈ T , then γ
s+t 2
=
1 1 (γ(s) + γ(t)) = (sv + tv) = 2 2
s+t 2
v.
Thus T is closed under taking midpoints. By Lemma 4.2 T = [0, 1] since T is easily seen to be closed. If b = 1 then we first scale by b from [0, 1] to [0, b] and apply the preceding paragraph to the composition of γ with the scaling. 2 Theorem 4.4. Let X, Y be manifolds with sprays and let f : X → Y be a continuous and locally midpointpreserving map. Then f is smooth. Furthermore, if the exponential maps for X, Y are analytic, then f is analytic. Proof. We choose a distinguished point a ∈ X, take f (a) for the distinguished point in Y , and define g locally near 0a ∈ Ta X by g = logf (a) ◦f ◦ expa .
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f
X
Y
expf (a)
expa
Ta X
139
g
Tf (a) Y
Here “locally” means choosing a normal ball B = expa (B(0a , r)) such that f (B) is contained in a normal ball around f (a) and so that log can be defined as the inverse of expf (a) . In this proof we denote {tu : 0 ≤ t ≤ 1} = [0, 1]u and {expa (tu) : 0 ≤ t ≤ 1} = expa ([0, 1]u) for convenience. We proceed in steps. Step 1. We show for u ∈ B(0a , r) there exists w ∈ Tf (a) Y such that g(tu) = tw for all t ∈ [0, 1]. From Lemma 3.6 we have seen the mapping t → expa (tu) is midpoint-preserving, and thus so is tu → expa (tu) from [0, 1]u to expa ([0, 1]u). Let x = expa (u), let y = f (x), and let v = logf (a) (y). Then expf (a) ([0, 1]v) is also closed under the midpoint operation #, and logf (a) is midpoint-preserving from expf (a) ([0, 1]v) to [0, 1]v. The map t → tu → expa (tu) → f (expa (tu)) from [0, 1] into Y is a composition of midpoint-preserving maps, and hence is midpoint-preserving. Since it carries 0a to f (a) and 1 to f (x) = y = expf (a) (v), one argues directly from Lemma 4.2 that all dyadic rational numbers are carried into expf (a) ([0, 1]v), and hence all of [0, 1] are since expf (a) ([0, 1]v) is compact and the composition is continuous. It follows that f (expa ([0, 1]u)) ⊆ expf (a) ([0, 1]v). Combining all this information together, we conclude that g = logf (a) ◦f ◦ expa restricted to [0, 1]u is midpoint-preserving. Thus t → g(tu) is a continuous and midpoint-preserving map carrying 0 to 0f (a) . By Lemma 4.3 there exists w ∈ Tf (a) Y such that g(tu) = tw for all t ∈ [0, 1]. Step 2. We show that we can uniquely u extend g so that g is globally defined, continuous, and positively for any real number m > 0. Let u ∈ Ta X and choose 0 < m < n homogeneous, that is, g(u) = mg m m u u u u < 1. By Step 1, there exists w ∈ Tf (a) Y such such that , ∈ B(0a , r). Then = s , where s = n m n um n that g t = tw for all t ∈ [0, 1]. It follows that m u m su m u ng = g = sw = mw = mg . n s m s m u Thus the definition of g(u) = mg is independent of the real number m > 0, as long as m is sufficiently m u large so that < r. m u For u ∈ Ta X, fix m > 0 such that < r. Then there is an open neighborhood U of u such that m v v , which is continuous. < r for v ∈ U . Then on U the extension of g is given by v → mg m m su For u ∈ Ta X and s > 0, pick m > s such that < r. Then m s su =m g(u) = sg(u). g(su) = mg m m Thus g is positively homogeneous.
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Step 3. We use the Lie–Trotter formula to show that g is additive. We temporarily abbreviate the exponential functions expa and expf (a) by exp, distinguishing them by context, and the corresponding log functions by log. We first use the positive homogeneity and the local equality g = log ◦f ◦ exp in various equivalent forms to calculate for u, v ∈ Ta X and n large: 2u 2v g n log exp # exp n n 2u 2v = n(g ◦ log) exp # exp n n 2v 2u # exp = n(log ◦f ) exp n n 2u 2v = n log f exp #f exp n n 2u 2v = n log exp g # exp g n n 2 2 g(u) # exp g(v) . = n log exp n n We thus have by the Lie–Trotter formula: g(u + v) = g
lim n log exp
n→∞
2u n
# exp
2v n
2u 2v = lim g n log exp # exp n→∞ n n 2 2 g(u) # exp g(v) = lim n log exp n→∞ n n = g(u) + g(v). Step 4. Since g is continuous, additive, and positively homogeneous, it follows easily that it is a continuous linear mapping, hence smooth. Near a, f = expf (a) ◦g ◦ loga and is thus smooth. Since a was an arbitrary choice for the distinguished point, the map f is smooth in a neighborhood of every point, hence smooth. If each exponential map expx is analytic, then the local inverse log is also analytic since the derivative at 0x is invertible (indeed, it is identity). Thus, the composition f = expf (x) ◦g ◦ logx is locally analytic, hence analytic. 2 5. Midpoints in symmetric spaces In this section we consider symmetric spaces in the sense of Loos, review some pertinent results, and then draw connections with the previous material on midpoints. Definition 5.1. We say (X, •) is a Loos symmetric space if X is a smooth Banach manifold, and (x, y) → x • y : X × X → X is a smooth map satisfying for all a, b, c ∈ X: (S1) (S2) (S3) (S4)
a • a = a; a • (a • b) = b; a • (b • c) = (a • b) • (a • c); Every a ∈ X has a neighborhood U such that a • x = x implies a = x for all x ∈ U .
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We view the left translation Sx (y) = x • y as a point reflection or symmetry through x. Properties (S1) and (S2) have obvious geometric interpretation. Remark 5.2. Geometric intuition suggests that reflection through the midpoint m of a and b should carry a to b. In other words, the solution of Sx a = x • a = b should be x = m, provided the midpoint m exists and is unique. Loos symmetric spaces on Banach manifolds have been recently studied by K.-H. Neeb [6], and Lawson and Lim [4]. Let us review some of their results. Theorem 5.3. (See [6].) Let (X, •) be a Loos symmetric space. (i) Identifying T (X × X) with T (X) × T (X), then v • w := T (μ)(v, w) where μ(x, y) := x • y defines a Loos symmetric space on T (X). In each tangent space Tx X, v • w = 2v − w. (ii) The function F : T (X) → T (TX ),
(iii) (iv) (v) (vi)
F (v) := −T (Sv/2 ◦ Z)(v)
defines a spray on X, where Z : X → TX is the zero section and Sv/2 is the point symmetry for v/2 from part (i). Aut(X, •) = Aut(X, F ), where the former consists of all diffeomorphisms that are automorphisms with respect to • and the latter consists of all diffeomorphisms that preserve the spray F . F is uniquely defined as the only spray invariant under all symmetries Sx , x ∈ X. Every geodesic of (X, F ) extends to a geodesic defined on all of R. Let α : R → X be a geodesic and call the maps τα,s := Sα(s/2) ◦ Sα(0) , s ∈ R, translations along α. Then these are automorphisms of (X, •) with τα,s (α(t)) = α(t + s)
and
dτα,s (α(t)) = Ptt+s (α)
for all s, t ∈ R. We write Ptt+s (α) : Tα(t) X → Tα(t+s) X for the linear map given by parallel transport along α. The set of real numbers R endowed with the core operation s • t = s + (−t) + s = 2s − t defines a Loos symmetric space. Proposition 5.4. (See [4].) The (maximal) geodesics of a Loos symmetric space (X, •) equipped with its unique symmetry-invariant spray are precisely the continuous homomorphisms β : (R, •) → (X, •), which are each of the form βv (t) = expx (tv), where x ∈ X, v ∈ Tx X, and hence are smooth. The correspondence v ↔ βv is a one-to-one correspondence between Tx X and all maximal geodesics taking on the value x at 0. We recall some examples of sprays on Loos symmetric spaces.
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Example 5.5. Let E be a Banach space with trivial tangent bundle E ×E. Consider that (E, •) is a symmetric space with x •y = 2x −y. Then the tangent bundle T (E) becomes a Loos symmetric space by Theorem 5.3(1). If we denote a point in the tangent bundle by a pair (x, v) in E × E, then we have
S(x, v2 ) ◦ Z (y) = S(x, v2 ) (y, 0) = (2x − y, v),
and so, T (S(x, v2 ) ◦ Z)(y, w) = (2x − y, v, −w, 0) ∈ (X × E) × (E × E). Therefore, F (x, v) = −T (S(x, v2 ) ◦ Z)(x, v) = (x, v, v, 0). Example 5.6. The open convex cone Ω of all positive definite operators on a Hilbert space is a Loos symmetric space with respect to the core operation μ(A, B) := A • B = AB −1 A. Then the tangent bundle T Ω can be denoted by T Ω = Ω × Herm, where Herm is the space of hermitian operators. Moreover, (T Ω, T μ) becomes a Loos symmetric space, where T μ((A, X), (B, Y )) = (A • B, XB −1 A + AB −1 X − AB −1 Y B −1 A). This implies that
1 S(A, X ) ◦ Z (B) = S(A, X ) (B, 0) = A • B, (XB −1 A + AB −1 X) . 2 2 2
By [6, Lemma 3.2] the first coordinate of the differential d(S(A, X ) ◦ Z)(A) is −idTA (Ω) , and the second 2 coordinate at X is 1 (−XB −1 XB −1 A − AB −1 XB −1 X). 2 So 1 −1 −1 T (S(A, X ) ◦ Z)(A, X) = A • A, X, −X, (−XA X − XA X) 2 2 = (A, X, −X, −XA−1 X). Therefore, the canonical spray is given by F (A, X) = −T (S(A, X ) ◦ Z)(A, X) = (A, X, X, XA−1 X). 2
On a Loos symmetric space, we can define the midpoint locally (as in Section 3) by means of the exponential map associated with the canonical spray. In the next proposition we show that this midpoint can be alternatively characterized as a midpoint of symmetry, i.e., a point m = m(x, y) that satisfies Sm x = m • x = y and Sm y = m • y = x.
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Proposition 5.7. Let (X, •) be a Loos symmetric space. Then the midpoint locally defined in DW can be alternatively characterized as the unique midpoint of symmetry in W . Proof. Let x0 ∈ X and let (W, V ) be a normal neighborhood pair for x0 (Theorem 2.12). For any pair (x, y) ∈ DW , we have a geodesic βv (t) = expx (tv) with βv (0) = x, βv (1) = y and βv ([0,1]) ⊆ W . According 1 v . Furthermore, to Definition 3.2, the midpoint m of x and y is given by m = x#y = βv (1/2) = expx 2 by Proposition 5.4 y = βv (1) = βv (2(1/2) − 0) = βv
1 •0 2
1 = βv • βv (0) = m • x. 2
Suppose m ∈ W also satisfies y = m • x. By Theorem 2.12(2)(ii), the exponential expm maps the open subset {m } × B() of Tm X diffeomorphically onto an open set U (m ) containing W and contained in U0 . Let expm (−w) = x and set βw (t) = expm (tw). By Proposition 5.4, βw is defined on all of R and is a homomorphism with respect to the operation •. Thus βw (1) = βw (0 • −1) = βw (0) • βw (−1) = m • x = y. Setting γ(t) = βw (2t −1), we obtain a geodesic on [0, 1] running from x to y (Remark 2.8), and by uniqueness of geodesics in U0 , we obtain γ(t) = x#t y for 0 ≤ t ≤ 1, in particular, x#y = γ(1/2) = βw (0) = m . 2 It follows immediately from Proposition 5.7 that in a smooth Loos symmetric space the operations • and # locally determine each other in the sense that locally m •x = y if and only if m = x#y. Thus a continuous map between two such symmetric spaces locally preserves one of the operations (see Definition 4.1) if and only if it locally preserves the other. We thus have the following corollary to Theorem 4.4. Corollary 5.8. Let (X, •) and (Y, •) be smooth Loos symmetric spaces, and let h : X → Y be a continuous map that is locally a homomorphism with respect to • at each point of X. Then h is smooth. The conclusion holds in particular if h is a continuous •-homomorphism. 6. Midpoints in Riemannian manifolds In this section we review how to construct unique local midpoints from the distance function associated with a smooth Riemannian manifold modeled on a finite or infinite dimensional Hilbert space. Let (X, g) be a connected Riemannian manifold, a smooth connected manifold X with a smooth section g of positive-definite quadratic forms on the tangent bundle giving each tangent space the structure of a Hilbert space. We recall some fundamental results about Riemannian manifolds, see Chapters VII–IX in [2]. For any piecewise smooth curve α : [a, b] → X, an arc length L(α) is given by b L(α) =
α (t) dt,
a
where α (t) = α (t), α (t)α(t) . The g-distance or simply distance δ is defined by setting δ(x, y) equal to the arc length infimum over all piecewise smooth curves from x to y. The distance function δ is indeed a metric, and the corresponding metric topology agrees with the original topology on X (see Proposition 6.1 in [2, Chapter VII]). Any Riemannian manifold X has a unique connection associated with the metric g, called the Levi-Civita connection or the canonical connection. This is typically used to define the geodesics of the Riemannian 1/2
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manifold, although in [2, Section VII.7] S. Lang has defined them alternatively (and equivalently) from the canonical spray associated to a Riemannian structure. The map t → exp(tv) for tv ∈ D, where D is the domain of the exponential map as in Section 2, gives the unique maximal geodesic that passes through x = exp(0x ) at t = 0 with velocity v ∈ Tx X. For v ∈ Tx X the geodesic t → exp(tv) has constant speed v , and hence for v ∈ D, this geodesic restricted to [0, 1] has length v . A geodesic β : [a, b] → X is called a minimal geodesic if L(β) ≤ L(α) for every piecewise smooth curve α joining β(a) = x to β(b) = y. This means equivalently that L(β) = δ(x, y). Since the geodesic β has some constant speed σ, we have δ(x, y) = L(β) = σ(b − a). It is straightforward to see that the restriction of β to any closed subinterval of [a, b] will again be a minimal geodesic. An open set U of X is said to be convex if given x, y ∈ U there exists a unique minimal geodesic in U joining x and y, where uniqueness means unique up to reparameterization. The following is a form of Whitehead’s theorem (see Theorem 5.8 and Corollary 5.3 in [2, Chapter VII]). Theorem 6.1. Let (X, g) be a Riemannian manifold, and let p ∈ X, Then there exists > 0 such that the following are satisfied: (1) For 0 < r ≤ , the open neighborhood B(p, r) = expp Bg (0p , r) is convex, where the second ball is taken in Tp X in the norm arising from g. (2) The map expp restricted to Bg (0p , ) is a diffeomorphism. (3) For 0 < r ≤ , the mapping (x, y) → v, where v ∈ Tx X and t → expx (tv) for 0 ≤ t ≤ 1 is the minimal geodesic joining x and y, is a diffeomorphism from B(p, r) × B(p, r) onto an open subset of D, the domain of exp. (4) For any v ∈ Bg (0p , ), the mapping tv → exp(tv) for 0 ≤ t ≤ 1 is an isometry. (5) For 0 < r ≤ /2 and x, y ∈ B(p, r), there is exactly one metric midpoint (with respect to the distance δ) between x and y. For metric spaces with distance δ, a point m is said to be a (metric) midpoint of x and y if δ(x, m) = δ(m, y) =
1 δ(x, y). 2
In general points x and y may have no, one, or several midpoints. However, there is only one in the case that there exists a unique minimal geodesic connecting x to y. Indeed, let β be a parameterization of this minimal geodesic with domain [a, b] and constant speed σ, where β(a) = x and β(b) = y. Then for c = (a + b)/2 and m = β(c), δ(x, m) = L(β|[a,c] ) = σ(c − a) =
1 1 σ(b − a) = δ(x, y), 2 2
where m = β(c). Similarly δ(m, y) = (1/2)δ(x, y), and no other point on the geodesic is a midpoint. We denote this uniquely determined midpoint for pairs (x, y) that lie on a unique minimal geodesic by x#y. In the next proposition we show that this midpoint can be characterized as a midpoint locally defined by the exponential map associated with the canonical spray. Proposition 6.2. Let (X, g) be a connected Riemannian manifold. Then the geodesic midpoint locally defined in DW (from the associated spray or Levi-Civita connection) can be characterized as the unique local metric midpoint in terms of Riemannian distance δ.
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Proof. For any pair (x, y) ∈ DW let β(t) = expx (tv), 0 ≤ t ≤ 1, be a geodesic with β(0) = x and β(1) = y. By Theorem 6.1(3) the map β is the unique minimal geodesic joining x and y. Hence by our previous discussion x#y is the image of geodesic β at t = 1/2, which means it agrees with the midpoint locally defined by the exponential map associated with the canonical spray. 2 Combining together Proposition 6.2 and Theorem 4.4, we obtain the following corollary; see Theorem 4.3 of [3]. Corollary 6.3. Let X, Y be smooth Riemannian manifolds and let σ : X → Y be a continuous and locally midpoint preserving map. Then σ is smooth. Furthermore, if the exponential maps for X, Y are analytic, then σ is analytic. We close by returning to a special case of Example 5.6. Example 6.4. We consider the manifold M of n × n-positive definite Hermitian matrices. There is a natural and well-known Riemannian metric, or called the Riemannian trace metric, on M making it a Riemannian manifold, indeed a symmetric space, of negative curvature (see, e.g., [5]). The corresponding distance trace n 1 metric between two positive definite matrices is given by δ(A, B) = ( i=1 log2 λi (A−1 B)) 2 , where λi (X) denotes the ith eigenvalue of X in non-decreasing order. By [5] the metric midpoint (with respect to the distance trace metric) agrees with the symmetric midpoint, which is the solution X of XA−1 X = X • A = B (see Example 5.6). By elementary algebra, one sees that the unique positive definite solution is given by A#B = A1/2 (A−1/2 BA−1/2 )1/2 A1/2 , the geometric mean of the two positive definite matrices A, B. The local validity of Proposition 5.7 extends to the global case in this setting (two distinct points uniquely determine a globally defined geodesic), so that the midpoints are also the geodesic midpoints for the canonical spray associated with the symmetric space structure. Hence our three methods for defining midpoints (geodesic, symmetric, and metric) all converge. 7. Closing comments The material presented in this paper applies also to certain settings in Finsler geometry, although we have not as of yet systemically worked out the details nor the range of applications. What one needs are a Banach manifold M equipped with both a spray and a Finsler structure that are appropriately interrated in the following sense: • parallel transport along the geodesics of the spray are isometries of the tangent spaces equipped with their Finsler norms; • each point of M has a neighborhood for which the geodesics of the spray are minimal geodesics for the Finsler metric, i.e., the Finsler distance between two points x, y in the neighborhood is the Finsler length of the unique geodesic of the spray contained in the neighborhood (see Theorem 2.12). (We remark however that even locally the minimal geodesics for the Finsler spray need not be unique – we are only requiring that the spray geodesic be one of them.) One can generalize much the preceding material on Riemannian manifolds to this setting. One shows that in a neighborhood Nx of 0x , x ∈ M , the map t → expx (tv), v ∈ Nx , from [0, 1] into Nx is an isometry and so (1/2)v maps to the metric midpoint of the image. It follows that locally midpoints with respect to the spray are also midpoints with respect to the Finsler metric, a weakened version of Proposition 6.2.
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Specific settings where the previous conditions are satisfied include Finsler symmetric spaces of nonpositive curvature modeled on general Banach spaces including the open cone of positive invertible elements for a C ∗ -algebra [6], [4, Sections 8,9] and the Hilbert metric on an open bounded convex subset of Rn [8]. References [1] [2] [3] [4] [5] [6] [7] [8]
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