Vanishing S-curvature of Randers spaces

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1 Vanshng S-curvature of Randers spaces Shn-ch OHTA Department of Mathematcs, Faculty of Scence, Kyoto Unversty, Kyoto , JAPAN (e-mal: December 31, 010 Abstract We gve a necessary and suffcent condton on a Randers space for the exstence of a measure for whch Shen s S-curvature vanshes everywhere. Moreover, f t exsts, such a measure concdes wth the Busemann-Hausdorff measure up to a constant multplcaton. Keywords: Randers spaces, S-curvature, Rcc curvature Mathematcs Subject Classfcaton (000): 53C60 1 Introducton Ths short artcle s concerned wth a characterzaton of Randers spaces admttng measures wth vanshng S-curvature. A Randers space (due to Randers [Ra]) s a specal knd of Fnsler manfold (M, F ) whose Fnsler structure F : T M [0, ) s wrtten as F (v) = α(v) + β(v), where α s a norm nduced from a Remannan metrc on M and β s a one-form on M. Randers spaces are mportant n applcatons and reasonable for concrete calculatons. See [AIM] and [BCS, Chapter 11] for more on Randers spaces. We equp a Fnsler manfold (M, F ) wth an arbtrary smooth measure m. Then the S-curvature S(v) R of v T M ntroduced by Shen (see [Sh, 7.3]) measures the dfference between m and the volume measure of the Remannan structure nduced from the tangent vector feld of the geodesc η wth η(0) = v (see. for the precse defnton). The author s recent work [Oh], [OS] on the weghted Rcc curvature (n connecton wth optmal transport theory) shed new lght on the mportance of ths quantty. A natural and mportant queston arsng from the theory of weghted Rcc curvature s: when does (M, F ) admt a measure m wth S 0? If such a measure exsts, then we can choose t as a good reference measure. Our man result provdes a complete answer to ths queston for Randers spaces. Theorem 1.1 A Randers space (M, F ) admts a measure m wth S 0 f and only f β s a Kllng form of constant length. Moreover, then m concdes wth the Busemann- Hausdorff measure up to a constant multplcaton. Supported n part by the Grant-n-Ad for Young Scentsts (B)

2 It has been observed by Shen [Sh, Example 7.3.1] that a Randers space wth the Busemann-Hausdorff measure satsfes S 0 f β s a Kllng form of constant length. Our theorem asserts that hs condton on β s also necessary for the exstence of m wth S 0, and then t mmedately follows that m must be a constant multplcaton of the Busemann-Hausdorff measure. On the one hand, Shen s result (the f part of Theorem 1.1) ensures that there s a rch class of non-remannan Randers spaces satsfyng S 0. On the other hand, the only f part says that the class of general Randers spaces s much wder and many Randers spaces have no measures wth S 0. Ths means that there are no canoncal (reference) measures on such Fnsler manfolds (n respect of the weghted Rcc curvature). Therefore, for a general Fnsler manfold, t s natural to start wth an arbtrary measure, as was dscussed n [Oh] and [OS]. Prelmnares for Fnsler geometry We frst revew the bascs of Fnsler geometry. Standard references are [BCS] and [Sh]. We wll follow the notatons n [BCS] wth a lttle change (e.g., we use v nstead of y )..1 Fnsler structures Let M be a connected n-dmensonal C -manfold wth n, and π : T M M be the natural projecton. Gven a local coordnate (x ) n =1 : U R n of an open set U M, we wll always denote by (x ; v ) n =1 the local coordnate of π 1 (U) gven by v = v ( / x ) π(v). A C -Fnsler structure s a functon F : T M [0, ) satsfyng the followng condtons: (I) F s C on T M \ {0}; (II) F (cv) = cf (v) for all v T M and c 0; (III) The n n matrx g j (v) := 1 (F ) v v (v) j s postve-defnte for all v T M \ {0}. The postve-defnte matrx (g j (v)) defnes a Remannan structure g v of T x M through ( g v a x, ) b j := g x j j (v)a b j. (.1) j,j Note that g v (v, v) = F (v). Ths nner product g v s regarded as the best approxmaton of F Tx M n the drecton v. Indeed, the unt sphere of g v s tangent to that of F Tx M at v/f (v) up to the second order. If (M, F ) s Remannan, then g v always concdes wth the orgnal Remannan metrc. As usual, (g j ) wll stand for the nverse matrx of (g j ).

3 We defne the Cartan tensor A jk (v) := F (v) g j v (v) k for v T M \ {0}, and remark that A jk 0 holds f and only f (M, F ) s Remannan. We also defne the formal Chrstoffel symbol γ jk(v) := 1 { g l glj (v) x (v) + g kl k x (v) g } jk j x (v) l l for v T M \{0}. Then the geodesc equaton s wrtten as η +G( η) = 0 wth the geodesc spray coeffcents G (v) := γ jk(v)v j v k j,k for v T M (G (0) := 0 by conventon). Usng these, we further defne the nonlnear connecton N j(v) := { γ jk(v)v k 1 } F (v) A jk(v)g k (v) k for v T M (N j(0) := 0 by conventon), where A jk(v) := l gl (v)a ljk (v). Note that (see [BCS, Exercse.3.3]) N j(v) = 1 G v (v). j. S-curvature and weghted Rcc curvature We choose an arbtrary postve C -measure m on a Fnsler manfold (M, F ). Fx a unt vector v F 1 (1) and let η : ( ε, ε) M be the geodesc wth η(0) = v. Along η, the tangent vector feld η defnes the Remannan metrc g η va (.1). Denotng the volume form of g η by vol η, we decompose m nto m(dx) = e Ψ( η) vol η (dx) along η. Then we defne the S-curvature of v by d(ψ η) S(v) := (0). dt We extend ths defnton to all w = cv wth c 0 by S(w) := cs(v). Clearly S 0 holds on Remannan manfolds wth the volume measure. The weghted Rcc curvature s defned n a smlar manner as follows: () Rc n (v) := Rc(v) + (Ψ η) (0) f S(v) = 0, Rc n (v) := otherwse; () Rc N (v) := Rc(v) + (Ψ η) (0) S(v) /(N n) for N (n, ); () Rc (v) := Rc(v) + (Ψ η) (0). Here Rc(v) s the usual (unweghted) Rcc curvature of v. The author [Oh] shows that boundng Rc N from below by K R s equvalent to the curvature-dmenson condton CD(K, N), and then there are many analytc and geometrc applcatons. Observe that the bound Rc n K > makes sense only when the S-curvature vanshes everywhere. 3

4 Therefore the class of such specal trples (M, F, m) deserves a partcular nterest. We remark that, f there are two measures m 1, m on (M, F ) satsfyng S 0, then m 1 = c m holds for some postve constant c. We rewrte S(v) accordng to [Sh, 7.3] for ease of later calculaton. Recall that η s the geodesc wth η(0) = v. Fx a local coordnate (x ) n =1 contanng η and represent m along η as m(dx) = σ(η) dx 1 dx dx n = σ(η) det(g η ) vol η(dx). We have by defnton S(v) = d ( ) det(g η(t) ) log = dt t=0 σ(η(t)) 1 d [ det(g η(t) ) ] det(g v ) dt t=0 v σ σ(x) x (x). Snce η solves the geodesc equaton η + G( η) = 0, the frst term s equal to 1 { g j (v) g j x k (v)vk + g j (v) g } j v k (v) ηk (0),j,k = { γ k(v)v k 1 } F (v) A k(v)g k (v) = N (v).,k Thus we obtan S(v) = {N (v) } v σ σ(x) x (x). (.) Observe that S(cv) = cs(v) ndeed holds for c 0 n ths form..3 Busemann-Hausdorff measure and Berwald spaces Dfferent from the Remannan case, there are several constructve measures on a Fnsler manfold, each of them s canoncal n some sense and concdes wth the volume measure for Remannan manfolds. Among them, here we treat only the Busemann-Hausdorff measure whch s actually the Hausdorff measure assocated wth the sutable dstance structure f F s symmetrc n the sense that F ( v) = F (v) holds for all v T M. Roughly speakng, the Busemann-Hausdorff measure s the measure such that the volume of the unt ball of each tangent space equals the volume of the unt ball n R n. Precsely, usng a bass w 1, w,..., w n T x M and ts dual bass θ 1, θ,..., θ n Tx M, the Busemann-Hausdorff measure m BH (dx) = σ BH (x) θ 1 θ θ n s defned as ω n σ BH (x) = vol n ({ (c ) R n F ( ) }) c w < 1, where vol n s the Lebesgue measure and ω n s the volume of the unt ball n R n. Let (M, F ) be a Berwald space (see [BCS, Chapter 10] for the precse defnton). Then t s well known that S 0 holds for the Busemann-Hausdorff measure (see [Sh, Proposton 7.3.1]). In fact, along any geodesc η : [0, l] M, the parallel transport T 0,t : T η(0) M T η(t) M wth respect to g η preserves F. Therefore choosng parallel vector felds along η as a bass yelds that σ BH s constant on η, whch yelds S 0. 4

5 3 Proof of Theorem 1.1 Let (M, F ) be a Randers space,.e., F (v) = α(v) + β(v) such that α s a norm nduced from a Remannan metrc and that β s a one-form. In a local coordnate (x ) n =1, we can wrte α(v) = a j (x)v v j, β(v) = b (x)v,j for v T x M. The length of β at x s defned by β (x) :=,j aj (x)b (x)b j (x), whch s necessarly less than 1 n order to guarantee F > 0 on T M \ {0}. We denote the Chrstoffel symbol of (a j ) by γ jk. We also defne b (x) := j a j (x)b j (x), b j (x) := b x j (x) k b k (x) γ k j(x). Note that b j s the coeffcent of the covarant dervatve of β wth respect to α, namely / x jβ = b jdx. We fnd by calculaton that ( β ) (x) = x j b j (x)b j (x). (3.1) We say that β s a Kllng form f b j +b j 0 holds on M. The geodesc spray coeffcents of F are gven by (see [BCS, ( )]) G (v) = j,k = j,k γ jk(v)v j v k [ γ jk(x)v j v k + b j k (x) ( a j (x)v k a k (x)v j) α(v) + b j k (x) v { v j v k + ( b k (x)v j b j (x)v k) α(v) }] F (v) =: j,k γ jk(x)v j v k + X (v) + Y (v). (3.) If S 0 on T x M, then we deduce from (.) that N (v) s lnear n v T x M. We shall see that only ths nfntesmal constrant s enough to mply the condton on β stated n Theorem 1.1. To see ths, we calculate N = G / v usng (3.). As the frst term j,k γ jk(x)v j v k comes from a Remannan structure, t suffces to consder only the lnearly of { X / v (v) + Y / v (v)}. For the sake of smplcty, we wll omt evaluatons at x and v n the followng calculatons. We frst obtan X = (b v j b j )a j α + b j k (a j v k a k v j ) a lv l α,j =,j,j,k,l b j (a j a j )α + j,k b j k (v k v j v j v k )α 1 = 0. 5

6 As Euler s theorem [BCS, Theorem 1..1] ensures ( ) v = 1 ( F v F ) = n 1 v F F v F, we next observe Y v = { v (b j + b j )v j + (b j b j )b j α + F,j k,l + n 1 { b j k v j v k + (b k v j b j v k )α } F = n + 1 j,k (b j + b j ) v v j F + (n + 1),j,j b j k (b k v j b j v k ) a } lv l (b j b j )b j αv F. By comparng the evaluatons at v and v, the coeffcents b j + b j n the frst term must vansh for all, j, and hence β s a Kllng form. For the second term, we fnd that (α/f ) j (b j b j )b j must be constant on each T x M. If α/f s not constant on some T x M (.e., β (x) 0), then t holds that j (b j b j )b j = 0. Snce β s a Kllng form, we deduce from (3.1) that α 0 = j (b j b j )b j = j b j b j = ( β ) x. Therefore β has a constant length as requred, for β = 0 s an open condton. If α/f s constant on some T x M, then the above argument yelds that β 0 on M. Ths completes the proof of the only f part of Theorem 1.1. For the f part, t s suffcent to show that the Busemann-Hausdorff measure satsfes S 0, that can be found n [Sh, Example 7.3.1]. We brefly repeat hs dscusson for completeness. We frst observe from [Sh, (.10)] that m BH (dx) = ( 1 β (x) ) (n+1)/ det(a j (x)) dx 1 dx n =: σ BH (x) dx 1 dx n. Snce β has a constant length, we have k v k σ BH σ BH (x) x (x) = 1 k,j,k v k a j (x) a j x k (x) =,j γ j(x)v j. Therefore we conclude, by (.), S(v) = 1,j,k [ ] γ jk(x)v j v k v k v k σ BH (x) σ BH (x) = 0. xk We fnally remark related known results and several consequences of Theorem

7 Remark 3.1 (a) A Randers space s a Berwald space f and only f β s parallel n the sense that b j 0 for all, j (see [BCS, Theorem ]). Thanks to [Sh, Example 7.3.], we know that a Kllng form of constant length s not necessarly parallel. (b) In [De], Deng gves a characterzaton of vanshng S-curvature for homogeneous Randers spaces endowed wth the Busemann-Hausdorff measure. (c) It s easy to construct a Randers space whose β does not have a constant length. Hence many Fnsler manfolds do not admt measures wth S 0 (n other words, wth Rc n K > ). (d) Another consequence of Theorem 1.1 s that only (constant multplcatons of) the Busemann-Hausdorff measures can satsfy S 0 on Randers spaces. Then a natural queston s the followng: Queston Is there a Fnsler manfold (M, F ) on whch some measure m other than (a constant multplcaton of) the Busemann-Hausdorff measure satsfes S 0? If yes, what knd of measure s m? If such a measure exsts, then t s more natural than the Busemann-Hausdorff measure n respect of the weghted Rcc curvature. References [AIM] P. L. Antonell, R. S. Ingarden and M. Matsumoto, The theory of sprays and Fnsler spaces wth applcatons n physcs and bology, Kluwer Academc Publshers Group, Dordrecht, [BCS] D. Bao, S.-S. Chern and Z. Shen, An ntroducton to Remann-Fnsler geometry, Sprnger- Verlag, New York, 000. [De] S. Deng, The S-curvature of homogeneous Randers spaces, Dfferental Geom. Appl. 7 (009), [Oh] S. Ohta, Fnsler nterpolaton nequaltes, Calc. Var. Partal Dfferental Equatons 36 (009), [OS] S. Ohta and K.-T. Sturm, Heat flow on Fnsler manfolds, Comm. Pure Appl. Math. 6 (009), [Ra] G. Randers, On an asymmetrcal metrc n the fourspace of general relatvty, Phys. Rev. () 59 (1941), [Sh] Z. Shen, Lectures on Fnsler geometry, World Scentfc Publshng Co., Sngapore,