Vanishing S-curvature of Randers spaces
Shin-ichi OHTA∗
Department of Mathematics, Faculty of Science, Kyoto University,
Kyoto 606-8502, JAPAN (e-mail: sohta@math.kyoto-u.ac.jp)

December 31, 2010

Abstract
We give a necessary and suﬃcient condition on a Randers space for the existenceof a measure for which Shen’s S-curvature vanishes everywhere. Moreover, if itexists, such a measure coincides with the Busemann-Hausdorﬀ measure up to a
constant multiplication.
Keywords: Randers spaces, S-curvature, Ricci curvature
Mathematics Subject Classiﬁcation (2000): 53C601Introduction

This short article is concerned with a characterization of Randers spaces admitting measures with vanishing S-curvature. A Randers space (due to Randers [Ra]) is a special
kind of Finsler manifold (M, F ) whose Finsler structure F : T M −→ [0, ∞) is written
as F (v) = α(v) + β(v), where α is a norm induced from a Riemannian metric on M andβ is a one-form on M . Randers spaces are important in applications and reasonable forconcrete calculations. See [AIM] and [BCS, Chapter 11] for more on Randers spaces.
We equip a Finsler manifold (M, F ) with an arbitrary smooth measure m. Thenthe S-curvature S(v) ∈ R of v ∈ T M introduced by Shen (see [Sh, §7.3]) measures thediﬀerence between m and the volume measure of the Riemannian structure induced fromthe tangent vector ﬁeld of the geodesic η with η̇(0) = v (see §2.2 for the precise deﬁnition).
The author’s recent work [Oh], [OS] on the weighted Ricci curvature (in connection withoptimal transport theory) shed new light on the importance of this quantity.
A natural and important question arising from the theory of weighted Ricci curvature
is: when does (M, F ) admit a measure m with S ≡ 0? If such a measure exists, then wecan choose it as a good reference measure. Our main result provides a complete answer
to this question for Randers spaces.
Theorem 1.1 A Randers space (M, F ) admits a measure m with S ≡ 0 if and only if
β is a Killing form of constant length. Moreover, then m coincides with the BusemannHausdorﬀ measure up to a constant multiplication.
∗Supported in part by the Grant-in-Aid for Young Scientists (B) 20740036.

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It has been observed by Shen [Sh, Example 7.3.1] that a Randers space with theBusemann-Hausdorﬀ measure satisﬁes S ≡ 0 if β is a Killing form of constant length.
Our theorem asserts that his condition on β is also necessary for the existence of m withS ≡ 0, and then it immediately follows that m must be a constant multiplication of theBusemann-Hausdorﬀ measure.
On the one hand, Shen’s result (the “if” part of Theorem 1.1) ensures that there isa rich class of non-Riemannian Randers spaces satisfying S ≡ 0. On the other hand,the “only if” part says that the class of general Randers spaces is much wider and manyRanders spaces have no measures with S ≡ 0. This means that there are no canonical(reference) measures on such Finsler manifolds (in respect of the weighted Ricci curvature). Therefore, for a general Finsler manifold, it is natural to start with an arbitrary
measure, as was discussed in [Oh] and [OS].

2Preliminaries for Finsler geometry

We ﬁrst review the basics of Finsler geometry. Standard references are [BCS] and [Sh].
We will follow the notations in [BCS] with a little change (e.g., we use v i instead of y i ).

2.1Finsler structures

Let M be a connected n-dimensional C ∞ -manifold with n ≥ 2, and π : T M −→ M
be the natural projection. Given a local coordinate (xi )ni=1 : U −→ Rn of an open set
U ⊂∑M , we will always denote by (xi ; v i )ni=1 the local coordinate of π −1 (U ) given byv = i v i (∂/∂xi )|π(v) .
A C ∞ -Finsler structure is a function F : T M −→ [0, ∞) satisfying the following
conditions:(I) F is C ∞ on T M \ {0};(II) F (cv) = cF (v) for all v ∈ T M and c ≥ 0;
(III) The n × n matrix
gij (v) :=1 ∂(F 2 )(v)
2 ∂v i ∂v j

is positive-deﬁnite for all v ∈ T M \ {0}.
The positive-deﬁnite matrix (gij (v)) deﬁnes a Riemannian structure gv of Tx M through(∑)∑∑
i ∂j ∂gva,b:=gij (v)ai bj .
(2.1)
ij
∂x
∂xiji,jNote that gv (v, v) = F (v)2 . This inner product gv is regarded as the best approximationof F |Tx M in the direction v. Indeed, the unit sphere of gv is tangent to that of F |Tx M atv/F (v) up to the second order. If (M, F ) is Riemannian, then gv always coincides withthe original Riemannian metric. As usual, (g ij ) will stand for the inverse matrix of (gij ).
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We deﬁne the Cartan tensor
Aijk (v) :=F (v) ∂gij(v)2 ∂v kfor v ∈ T M \ {0}, and remark that Aijk ≡ 0 holds if and only if (M, F ) is Riemannian.
We also deﬁne the formal Christoﬀel symbol{}
∂glj
∂gkl
∂gjk
1 ∑ ili(v) +(v) −(v)γ jk (v) :=g (v)
2 l
∂xk
∂xj
∂xlfor v ∈ T M \ {0}. Then the geodesic equation is written as η̈ + G(η̇) = 0 with the geodesic
spray coeﬃcients∑Gi (v) :=γ i jk (v)v j v kj,kfor v ∈ T M (Gi (0) := 0 by convention). Using these, we further deﬁne the nonlinearconnection}∑{
1ikiikγ jk (v)v −N j (v) :=A jk (v)G (v)F (v)k∑for v ∈ T M (N i j (0) := 0 by convention), where Ai jk (v) := l g il (v)Aljk (v). Note that(see [BCS, Exercise 2.3.3])
1 ∂Gi
N i j (v) =(v).
2 ∂v j2.2

S-curvature and weighted Ricci curvature

We choose an arbitrary positive C ∞ -measure m on a Finsler manifold (M, F ). Fix a unitvector v ∈ F −1 (1) and let η : (−ε, ε) −→ M be the geodesic with η̇(0) = v. Along η, thetangent vector ﬁeld η̇ deﬁnes the Riemannian metric gη̇ via (2.1). Denoting the volumeform of gη̇ by volη̇ , we decompose m into m(dx) = e−Ψ(η̇) volη̇ (dx) along η. Then we deﬁne
the S-curvature of v byd(Ψ ◦ η̇)S(v) :=(0).
dt
We extend this deﬁnition to all w = cv with c ≥ 0 by S(w) := cS(v). Clearly S ≡ 0 holds
on Riemannian manifolds with the volume measure.
The weighted Ricci curvature is deﬁned in a similar manner as follows:(i) Ricn (v) := Ric(v) + (Ψ ◦ η)00 (0) if S(v) = 0, Ricn (v) := −∞ otherwise;(ii) RicN (v) := Ric(v) + (Ψ ◦ η)00 (0) − S(v)2 /(N − n) for N ∈ (n, ∞);(iii) Ric∞ (v) := Ric(v) + (Ψ ◦ η)00 (0).
Here Ric(v) is the usual (unweighted) Ricci curvature of v. The author [Oh] shows thatbounding RicN from below by K ∈ R is equivalent to the curvature-dimension condition
CD(K, N ), and then there are many analytic and geometric applications. Observe thatthe bound Ricn ≥ K > −∞ makes sense only when the S-curvature vanishes everywhere.
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Therefore the class of such special triples (M, F, m) deserves a particular interest. We
remark that, if there are two measures m1 , m2 on (M, F ) satisfying S ≡ 0, then m1 = c·m2
holds for some positive constant c.
We rewrite S(v) according to [Sh, §7.3] for ease of later calculation. Recall that η isthe geodesic with η̇(0) = v. Fix a local coordinate (xi )ni=1 containing η and represent malong η asσ(η)
volη̇ (dx).
m(dx) = σ(η) dx1 dx2 · · · dxn = √det(gη̇ )We have by deﬁnition(√)] ∑ v i ∂σdet(gη̇(t) )1d ¯¯d ¯¯ [=det(g)−(x).
S(v) = ¯ log¯η̇(t)
idt t=0σ(η(t))
2 det(gv ) dt t=0σ(x)∂xiSince η solves the geodesic equation η̈ + G(η̇) = 0, the ﬁrst term is equal to{}∂gij
1 ∑ ij ∂gijkijkg (v) k (v)v + g (v) k (v)η̈ (0)2 i,j,k∂x∂v{} ∑∑1ikik
γ ik (v)v −=A ik (v)G (v) =N i i (v).
F (v)ii,kThus we obtain
S(v) =∑{i}v i ∂σ
N i (v) −(x) .
σ(x) ∂xii(2.2)

Observe that S(cv) = cS(v) indeed holds for c ≥ 0 in this form.

2.3

Busemann-Hausdorﬀ measure and Berwald spaces

Diﬀerent from the Riemannian case, there are several constructive measures on a Finsler
manifold, each of them is canonical in some sense and coincides with the volume measure
for Riemannian manifolds. Among them, here we treat only the Busemann-Hausdorﬀ
measure which is actually the Hausdorﬀ measure associated with the suitable distancestructure if F is symmetric in the sense that F (−v) = F (v) holds for all v ∈ T M .
Roughly speaking, the Busemann-Hausdorﬀ measure is the measure such that thevolume of the unit ball of each tangent space equals the volume of the unit ball in Rn .
Precisely, using a basis w1 , w2 , . . . , wn ∈ Tx M and its dual basis θ1 , θ2 , . . . , θ n ∈ Tx∗ M , theBusemann-Hausdorﬀ measure mBH (dx) = σBH (x) θ1 ∧ θ2 ∧ · · · ∧ θn is deﬁned as)})({¯ (∑ωniin¯c wi < 1 ,= voln(c ) ∈ R ¯ F
σBH (x)iwhere voln is the Lebesgue measure and ωn is the volume of the unit ball in Rn .
Let (M, F ) be a Berwald space (see [BCS, Chapter 10] for the precise deﬁnition).
Then it is well known that S ≡ 0 holds for the Busemann-Hausdorﬀ measure (see [Sh,Proposition 7.3.1]). In fact, along any geodesic η : [0, l] −→ M , the parallel transport
T0,t : Tη(0) M −→ Tη(t) M with respect to gη̇ preserves F . Therefore choosing parallel
vector ﬁelds along η as a basis yields that σBH is constant on η, which yields S ≡ 0.
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Proof of Theorem 1.1

Let (M, F ) be a Randers space, i.e., F (v) = α(v) + β(v) such that α is a norm induced
from a Riemannian metric and that β is a one-form. In a local coordinate (xi )ni=1 , we canwrite√∑∑α(v) =aij (x)v i v j ,β(v) =bi (x)v ii,jifor v ∈ Tx M . The length of β at x is deﬁned by kβk(x) :=√∑i,j aij (x)b (x)b (x), whichi
jis necessarily less than 1 in order to guarantee F > 0 on T M \ {0}.
We denote the Christoﬀel symbol of (aij ) by γ̃ i jk . We also deﬁnebi (x) :=∑aij (x)bj (x),bi|j (x) :=j∑∂bi(x)−bk (x)γ̃ k ij (x).
∂xjk

˜ of β with respect to α, namely
Note that bi|j is the coeﬃcient of the covariant derivative ∇∑i˜ ∂/∂xj β =∇i bi|j dx . We ﬁnd by calculation that∑∂(kβk2 )(x)=2bj|i (x)bj (x).
∂xij(3.1)

We say that β is a Killing form if bi|j +bj|i ≡ 0 holds on M . The geodesic spray coeﬃcientsof F are given by (see [BCS, (11.3.11)])∑γ i jk (v)v j v kGi (v) =j,k∑[()=γ̃ i jk (x)v j v k + bj|k (x) aij (x)v k − aik (x)v j α(v)j,k)}vi { j k ( k+ bj|k (x)v v + b (x)v j − bj (x)v k α(v)F (v)∑γ̃ i jk (x)v j v k + X i (v) + Y i (v).
=:]

(3.2)

j,k

∑
If S ≡ 0 on Tx M , then we deduce from (2.2) that i N i i (v) is linear in v ∈ Tx M .
We shall see that only this inﬁnitesimal constraint is enough to imply the condition on
β stated in
1.1. To see this, we calculate 2N i i = ∂Gi /∂v i using (3.2). As the∑Theoremﬁrst term j,k γ̃ i jk (x)v j v k comes from a Riemannian structure, it suﬃces to consider only∑
the linearly of i {∂X i /∂v i (v) + ∂Y i /∂v i (v)}. For the sake of simplicity, we will omit
evaluations at x and v in the following calculations.
We ﬁrst obtainl∑∑ ∂X i ∑ij
ij kik j ail v=(b−b)aα+b(av−av)
j|ii|jj|k∂v i
αi,jii,j,k,l∑∑=bj|i (aij − aji )α +bj|k (v k v j − v j v k )α−1 = 0.
i,jj,k5

As Euler’s theorem [BCS, Theorem 1.2.1] ensures()∑ ∂ ( vi )1 ∑n−1i ∂F= 2F −v i =,i∂v FF i∂v
Fi
we next observe}l∑ vi {∑jjk jj k ail v=(bi|j + bj|i )v + (bi|j − bj|i )b α +bj|k (b v − b v )∂v iFαi,jk,l{}n−1∑+bj|k v j v k + (bk v j − bj v k )αF j,k∑ ∂Y ii=∑n+1∑vivj
αv i(bi|j + bj|i )(bi|j − bj|i )bj+ (n + 1).
2
FFi,ji,j

By comparing the evaluations at v and −v, the coeﬃcients bi|j + bj|i in the ﬁrst term
must vanish
for all i, j, and hence β is a Killing form. For the second term, we ﬁnd that∑(α/F ) j (bi|j − bj|i )bj must be constant on each Tx M . If α/F is not constant on some∑Tx M (i.e., kβk(x) 6= 0), then it holds that j (bi|j − bj|i )bj = 0. Since β is a Killing form,
we deduce from (3.1) that0=∑(bi|j − bj|i )bj = −2j∑jbj|i bj = −∂(kβk2 ).
∂xi

Therefore β has a constant length as required, for kβk 6= 0 is an open condition. If α/F isconstant on some Tx M , then the above argument yields that β ≡ 0 on M . This completes
the proof of the “only if” part of Theorem 1.1.
For the “if” part, it is suﬃcient to show that the Busemann-Hausdorﬀ measure satisﬁes
S ≡ 0, that can be found in [Sh, Example 7.3.1]. We brieﬂy repeat his discussion forcompleteness. We ﬁrst observe from [Sh, (2.10)] that√()
2 (n+1)/2mBH (dx) = 1 − kβk(x)det(aij (x)) dx1 · · · dxn =: σBH (x) dx1 · · · dxn .
Since β has a constant length, we have∑k∑
1 ∑ k ij ∂aijv k ∂σBH(x)=(x)=γ̃ i ij (x)v j .
va(x)σBH (x) ∂xk
2 i,j,k∂xki,jTherefore we conclude, by (2.2),
S(v) =] ∑ v k ∂σ1∑ ∂ [ iBHj k−γ̃(x)vv(x) = 0.
jkk2 i,j,k ∂v iσ(x)∂xBHk2

We ﬁnally remark related known results and several consequences of Theorem 1.1.
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Remark 3.1 (a) A Randers space is a Berwald space if and only if β is parallel in thesense that bi|j ≡ 0 for all i, j (see [BCS, Theorem 11.5.1]). Thanks to [Sh, Example 7.3.2],
we know that a Killing form of constant length is not necessarily parallel.
(b) In [De], Deng gives a characterization of vanishing S-curvature for homogeneous
Randers spaces endowed with the Busemann-Hausdorﬀ measure.
(c) It is easy to construct a Randers space whose β does not have a constant length.
Hence many Finsler manifolds do not admit measures with S ≡ 0 (in other words, withRicn ≥ K > −∞).
(d) Another consequence of Theorem 1.1 is that only (constant multiplications of)
the Busemann-Hausdorﬀ measures can satisfy S ≡ 0 on Randers spaces. Then a natural
question is the following:
Question Is there a Finsler manifold (M, F ) on which some measure m other than (aconstant multiplication of) the Busemann-Hausdorﬀ measure satisﬁes S ≡ 0? If yes, whatkind of measure is m?
If such a measure exists, then it is more natural than the Busemann-Hausdorﬀ measure
in respect of the weighted Ricci curvature.

References[AIM] P. L. Antonelli, R. S. Ingarden and M. Matsumoto, The theory of sprays and Finsler
spaces with applications in physics and biology, Kluwer Academic Publishers Group,
Dordrecht, 1993.
[BCS] D. Bao, S.-S. Chern and Z. Shen, An introduction to Riemann-Finsler geometry, SpringerVerlag, New York, 2000.
[De]

S. Deng, The S-curvature of homogeneous Randers spaces, Diﬀerential Geom. Appl. 27(2009), 75–84.

[Oh]

S. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Diﬀerential Equations 36(2009), 211–249.

[OS]

S. Ohta and K.-T. Sturm, Heat ﬂow on Finsler manifolds, Comm. Pure Appl. Math. 62(2009), 1386–1433.

[Ra]

G. Randers, On an asymmetrical metric in the fourspace of general relativity, Phys. Rev.
(2) 59 (1941), 195–199.

[Sh]

Z. Shen, Lectures on Finsler geometry, World Scientiﬁc Publishing Co., Singapore, 2001.

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