Examples of spaces with branching geodesics
satisfying the curvature-dimension condition
Shin-ichi OHTA∗
July 17, 2013
Abstract
In this short note, we give two simple examples of metric measure spaces containing branching geodesics and satisfying Lott, Sturm and Villani’s curvaturedimension condition. In the ﬁrst example, only geodesics in a subspace of codimension 1 can branch. The second one is positively curved in the sense of thecurvature-dimension condition. We also discuss related open problems.

1Background

The aim of this half-expository note is to draw attention to examples of spaces satisfying
the curvature-dimension condition and containing branching geodesics. We ﬁrst explain
the background of this subject.

1.1

Curvature-dimension condition

The curvature-dimension condition for a (complete, separable) metric measure space(X, d, m) is a generalized notion of lower Ricci curvature bounds introduced independently by Sturm ([St1], [St2]) and Lott and Villani ([LV1], [LV2]). Given K ∈ R andN ∈ (1, ∞], a Riemannian manifold (M, g) equipped with the Riemannian volume measure satisﬁes the curvature-dimension condition CD(K, N ) if and only if Ricg ≥ K (i.e.,Ricg (v, v) ≥ Khv, vi for all v ∈ T M ) and dim M ≤ N . Metric measure spaces satisfying
CD(K, N ) are known to behave like enjoying ‘Ric ≥ K and dim ≤ N ’ in geometric andanalytic respects (see [St1], [St2], [LV1], [LV2] and [Vi, Part III]).
The curvature-dimension condition is a certain convexity condition of an entropy functional on the space of probability measures on X. Let us give the precise deﬁnition (in thesense of Sturm [St1]) only for N = ∞. Denote by P(X) the set of all Borel probability
measures on (X, d), and by P 2 (X) the subset consisting of µ ∈ P(X) with ﬁnite second
moment. Deﬁne the L2 -Wasserstein distance between µ, ν ∈ P 2 (X) by(∫)1/2
2W2 (µ, ν) := inf,d(x, y) π(dxdy)π∈Π(µ,ν)X×X∗

Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan (sohta@math.kyoto-u.ac.jp);
Supported in part by the Grant-in-Aid for Young Scientists (B) 23740048.

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where Π(µ, ν) is the set of all π ∈ P(X × X) whose ﬁrst and second projections to X
coincide with µ and ν, respectively. We call (P 2 (X), W2 ) the L2 -Wasserstein space over(X, d). The relative entropy Entm : P 2 (X) −→ [−∞, ∞] associated with a Borel measurem on X is deﬁned by∫
Entm (µ) :=ρ log ρ dm
X∫

if µ = ρm ¿ m and {ρ>1} ρ log ρ dm < ∞, otherwise we set Entm (µ) := ∞. Then(X, d, m) is said to satisfy CD(K, ∞) if any pair µ, ν ∈ P 2 (X) is connected by a minimal
geodesic (µt )t∈[0,1] ⊂ P 2 (X) (i.e., µ0 = µ, µ1 = ν, W2 (µs , µt ) = |s − t|W2 (µ, ν) for alls, t ∈ [0, 1]) satisfying
Entm (µt ) ≤ (1 − t) Entm (µ) + t Entm (ν) −K(1 − t)tW2 (µ, ν)22(1.1)

for all t ∈ [0, 1]. A minimal geodesic (µt )t∈[0,1] in (P 2 (X), W2 ) can be described via thepush-forward of µ0 along geodesics in X, so that the behavior of Entm (µt ) is naturally
controlled by the Ricci curvature in the Riemannian case. The deﬁnition of CD(K, N )for N < ∞ is more involved (especially for K 6= 0), here we only mention that CD(K, N )implies CD(K 0 , N 0 ) for any K 0 ∈ (−∞, K] and N 0 ∈ [N, ∞].

1.2

Branching geodesics

The equivalence between the curvature-dimension condition and lower Ricci curvaturebounds was extended to Finsler manifolds in [Oh3, Theorem 1.2]. Precisely, given an
n-dimensional Finsler manifold (M, F ) with a positive C ∞ -measure m, we consider theweighted Ricci curvature RicN for N ∈ [n, ∞] given byRicN (v) := Ric(v) + (ψ ◦ η)00 (0) −(ψ ◦ η)0 (0)2
N −nfor v ∈ Tx M,(1.2)

where Ric is the Finsler-Ricci curvature, η is the geodesic with η̇(0) = v, and ψ is determined by m = e−ψ volη̇ around x with the volume measure volη̇ of the Riemannian
metric gη̇ induced from η̇ (see [Oh3] for the precise deﬁnition). Then (M, F, m) satisﬁes
CD(K, N ) if and only if RicN (v) ≥ KF (v)2 for all v ∈ T M . By combining this equivalencewith the stability of CD(K, N ) under convergence of metric measure spaces, all normed
spaces (Rn , | · |) with the Lebesgue measure mL satisfy CD(0, n) (see also [Vi, Theorem inpage 908]). In particular, the `∞ -space (Rn , | · |∞ , mL ) satisﬁes CD(0, n).
Geodesics in (Rn , | · |∞ ) can branch in the sense that there exist two minimal geodesics
η1 , η2 : [0, 1] −→ Rn with respect to |·|∞ such that η1 (t) = η2 (t) on [0, ε] for some ε ∈ (0, 1)
and that η1 (t) 6= η2 (t) at some t ∈ (ε, 1]. The existence of branching geodesics causes
diﬃculties in the study of spaces satisfying the curvature-dimension condition. Although
the non-branching assumption has been recently removed in some cases (see [Ra1], [Ra2]),
there still remain important results for those we need to assume that geodesics do notbranch. See, for instances, [St1, Proposition 4.16] and [DS] for products of CD-spaces,and [Oh2, §5] for a maximal diameter rigidity. If geodesics do not branch (then (X, d) iscalled a non-branching space), then there is essentially only one minimal geodesic between
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absolutely continuous measures µ, ν ¿ m (see [St2, Lemma 4.1]), so that the entropy isconvex along every minimal geodesic. This enables us to localize CD(K, N ) to a certain
inequality (concavity of Jacobian) along almost all geodesics in X describing the minimalgeodesic between µ and ν (see [St2, Proposition 4.2]). Such an inﬁnitesimal inequality issometimes necessary for sharper estimates.
As far as the author knows, only known examples of spaces satisfying CD(K, N ) andcontaining branching geodesics are normed spaces. We shall present two more examplesof branching spaces (built as singular Finsler manifolds) in this note, whereas they arestill modiﬁcations of normed spaces in a sense. We also discuss open problems related tothese examples.

2An essentially non-branching example

2.1Construction

Given a bounded convex domain D ⊂ Rn , Hilbert [Hi] introduced the distance function( 0)|x − y| · |x − y 0 |
1dH (x, y) := log
for x, y ∈ D, x 6= y,2|x0 − x| · |y − y 0 |where | · | is the Euclidean norm, and x0 = x + s(y − x) and y 0 = x + t(y − x) are theintersections of the boundary ∂D and the line passing through x and y with s < 0 < t(see Figure 1). This is a generalization of the Klein model of a hyperbolic space whichcorresponds to the case where D is the unit ball. It is easily seen that, for any x, y ∈ D,the line segment η(t) := x + t(y − x) (t ∈ [0, 1]) gives a minimal geodesic from x to ywith respect to dH . If D is strictly convex (i.e., x, y ∈ ∂D implies (x + y)/2 ∈ D unlessx = y), then η is the unique minimal geodesic between x and y. Otherwise, minimalgeodesics may not be unique. For instance, consider the square D = (−1, 1)2 ⊂ R2 . SincedH ((0, 0), (a, b)) = dH ((0, 0), (a, b0 )) for any a ∈ (0, 1) and b, b0 ∈ [−a, a], we can constructuncountably many minimal geodesics between (0, 0) and (a, 0) with a > 0 (e.g., thickbroken segments in Figure 2).
y0byv Kxx0

aFigure 1

Figure 23

If ∂D is C ∞ and positively curved (the latter will be called the strong convexity of D),then dH is realized by the C ∞ -Finsler metric()
1|v|
1FH (x, v) :=+for v ∈ Tx D = Rn ,2 |x − a| |x − b|where a = x + sv and b = x + tv are the intersections of ∂D and the line passing through x
in the direction v with s < 0 < t (see Figure 1). It is well-known that the Finsler manifold(D, FH ) has the constant ﬂag curvature −1 (see [Sh, §12.2]). Recently, the author showedthat (D, FH ) equipped with the Lebesgue measure mL on D has the bounded weighted
Ricci curvature:(](](n + 1)2
Ric∞ (v) ∈ − (n − 1), 2 ,RicN (v) ∈ − (n − 1) −,2N −nfor any unit vector v ∈ T D and N ∈ (n, ∞) ([Oh4, Theorem 1.2]). Thus, for any bounded
convex domain D ⊂ Rn , (D, dH , mL ) satisﬁes CD(K, N ) with
K = −(n − 1) for N = ∞,K = −(n − 1) −(n + 1)2
N −nfor N ∈ (n, ∞)by approximating D by strongly convex domains with C ∞ -boundaries.
Now, let us consider a domain D ⊂ R3 such that P := D ∩ (R2 × {0}) = (−1, 1)2 × {0}and that ∂D \ (R2 × {0}) is C ∞ and positively curved. Then P contains branchinggeodesics, while geodesics not included in P do not branch. Indeed, for any x, y ∈ D withx 6∈ P or y 6∈ P , the line segment is a unique minimal geodesic between them. Hence(D, dH , mL ) is essentially non-branching in the sense of Theorem 2.2 below. A concrete
example of such a domain isD2 := {(a, b, c) ∈ R3 | (a, b) ∈ (−1, 1)2 , c2 < (1 − a2 )(1 − b2 )}.

(2.1)

Remark 2.1 In [Oh4], we also gave a lower bound of the weighted Ricci curvature for
Funk’s distance [Fu]:()|x − y 0 |dF (x, y) = log
for x, y ∈ D, x 6= y,|y − y 0 |which can be regarded as a ‘non-symmetrization’ of dH . In (D2 , dF ), however, geodesics
in a plane containing only one edge of ∂P (e.g., {1} × [−1, 1] × {0}) can branch. Therefore
there are many more branching geodesics than (D2 , dH ).

2.2

Open problems

The above example is inspired by the following recent work of Rajala and Sturm. Wesay that (X, d, m) satisﬁes the strong curvature-dimension condition sCD(K, ∞) if (1.1)
holds along any minimal geodesic in P 2 (X).
Theorem 2.2 ([RS, Corollary 1.2]) Let (X, d) be a complete separable metric space with alocally ﬁnite measure m on X, and suppose that (X, d, m) satisﬁes sCD(K, ∞). Then anyminimal geodesic between absolutely continuous measures µ0 , µ1 ∈ P 2 (X) is concentrated
on a set of non-branching geodesics.
4See [RS] for the precise statement. Our example (D2 , dH , mL ) in (2.1) satisﬁes thehypothesis of Theorem 2.2, thus it shows that geodesics still can branch in this kind of
essentially non-branching spaces.
An important class of spaces covered by Theorem 2.2 is the one of metric measurespaces enjoying the Riemannian curvature-dimension condition RCD(K, ∞) introducedby Ambrosio, Gigli and Savaré [AGS]. Roughly speaking, RCD(K, ∞) is deﬁned by the
combination of sCD(K, ∞) and the linearity of the heat ﬂow. The heat ﬂow on a Finsler
manifold (M, F, m) is linear if and only if F comes from a Riemannian metric (see [OS]),so that (D2 , dH , mL ) does not satisfy RCD(K, ∞) for any K.
Problem A Is there a metric measure space satisfying RCD(K, ∞) for some K ∈ R andcontaining branching geodesics?
The author does not have any idea how to construct such a space. If the answer isNO, then we can remove the non-branching assumption from a number of theorems onRCD-spaces. Furthermore, since all limit spaces of Riemannian manifolds with Ric ≥ Ksatisfy RCD(K, ∞), the negative answer to Problem A contributes to the developmentof the study of the local structure of these limit spaces. After seminal work of Cheegerand Colding [CC], it is the important open question whether such limit spaces admit
branching geodesics or not.
Problem B In the case where the answer to Problem A is YES, can one construct a
branching space satisfying RCD(K, ∞) as a limit space of Riemannian manifolds with
Ric ≥ K?
This is related to another deep question: Can any metric measure space satisfyingRCD(K, ∞) be realized as the limit of a sequence of Riemannian manifolds with Ric ≥ K(or weighted Riemannian manifolds with Ric∞ ≥ K)?

3A positively curved example

3.1Construction

We next construct a branching space satisfying CD(K, N ) with K > 0 and N < ∞. Fix
N > 2, R > 0 and a ∈ (0, (N − 2)R−2 ). Consider the metric measure space (XR , d∞ , ma )given by()a|x|222XR := {x ∈ R | |x|2 < R}, d∞ (x, y) := |x − y|∞ , ma := exp −mL (dx).
2Clearly geodesics can branch in (XR , d∞ ). Note that, along any line segment η in XR ,()0()00a|η|22a|η|22≤ a|η|2 · |η̇|2 ≤ aR|η̇|2 ,= a|η̇|22 .
22Since |η̇|2 ≥ |η̇|∞ , by approximating | · |∞ by C ∞ and strongly convex norms to be precise,we see that (XR , d∞ , ma ) satisﬁes CD(K, N ) with (recall (1.2))
K =a−5(aR)2.
N −2Note that K > 0 by the choice of a. We similarly ﬁnd that (R2 , d∞ , ma ) satisﬁes CD(a, ∞)for any a > 0.

3.2

Open problems

The above example is related to a rigidity result for positively curved spaces. By theBonnet–Myers type theorem, the diameter√ of a metric measure space satisfying CD(K, N )with K > 0 and N < ∞ is at most π (N − 1)/K. In [Oh2, §5], under the weaker assumption of the measure contraction property MCP(K, N ) (see√ [Oh1], [St2, §5]), we showedthat a non-branching space with the maximal diameter π (N − 1)/K is homeomorphic
to the spherical suspension of a topological measure space. The non-branching property
was necessary for proving the continuity of the homeomorphism.
In our example (XR , d∞ ), the diameter is 2R and is smaller than the maximal one:
√√
√
N −1
N −1
4R2
π=π≥
π(N−1)> 2πR.
Ka − (aR)2 /(N − 2)
N −2
Problem C Is there a metric measure√ space satisfying CD(K, N ) with K > 0 and N <∞, attaining the maximal diameter π (N − 1)/K, and containing branching geodesics?
If YES, can such a space be realized as the spherical suspension of some space?
One can weaken the condition CD(K, N ) to MCP(K, N ), or strengthen it to the combination of CD(K, N ) and RCD(K, ∞). Related to the latter, we can also ask the following.
Problem D Is there a metric measure space which satisﬁes CD(K, N ) with K > 0 (orK = 0), contains branching geodesics, but is essentially non-branching in the sense ofTheorem 2.2?
This would be a more accessible one than other problems. Recall that the space(D2 , dH , mL ) in the previous section admits only a negative curvature bound.
Acknowledgements. I would like to thank Professors Karl-Theodor Sturm and Tapio
Rajala for discussions during a workshop at Scuola Normale Superiore, Pisa, in November
2012. I am also grateful to the organizers of the workshop, Professors Luigi Ambrosio,
Giuseppe Buttazzo and Giuseppe Savaré, for their hospitality.

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