Table Of Content

UNIFORM BOUNDS FOR WEIL-PETERSSON CURVATURES 5 1 MICHAEL WOLF& YUNHUIWU 0 2 n Abstract. We find bounds for Weil-Petersson holomorphic sectional a curvature,andtheWeil-Peterssoncurvatureoperatorinseveralregimes, J that do not depend on the topology of the underlying surface. Among 9 other results, we show that the minimal (most negative) eigenvalue of 2 the curvatureoperator at any point in theTeichmu¨ller space Teich(Sg) of a closed surface Sg of genus g is uniformly bounded away from zero. G] RestrictingtoathickpartofTeich(Sg),weshowthattheminimaleigen- valueisuniformlyboundedbelowbyanexplicitconstantwhichdoesnot D depend on the topology of the surface but only on the given bound on . injectivity radius. We also show that the minimal Weil-Petersson holo- h morphic sectional curvature of a sufficiently thick hyperbolic surface is t a comparable to −1. m [ 2 1. Introduction v 5 Let S be a closed surface of genus g where g > 1, and Teich(S ) be the 4 g g 2 Teichmu¨ller space of Sg. Endowed with the Weil-Petersson metric, the Te- 3 ichmu¨ller space Teich(S ) is Ka¨hler ([1]), incomplete ([7, 33]), geodesically g 0 complete ([35]) and negatively curved ([30, 34]). Tromba [30] and Wolpert . 1 [34]foundaformulafortheWeil-Petersson curvaturetensor,whichhasbeen 0 applied to study a variety of curvature properties of Teich(S ) over the past 5 g 1 several decades. (See also [13, 16, 28] for alternative proofs of the curva- : ture formula.) In their papers, they deduced from their formula that the v i holomorphic sectional curvatures are bounded above by a negative number X which only depends on the genus of the surface, confirming a conjecture of r a Royden. We focus in this paper on bounds for Weil-Petersson curvatures that are uniform across Teichmu¨ller spaces, in the sense that the bounds do not depend on the genus g of the surface S . g Naturally, there are a number of appealing notions of curvature of a Rie- mannian manifold. In particular, while we focus on the holomorphic sectional curvatures in the second part of the paper, in the first part of this paper we study the Weil-Petersson curvature operator Q : 2T Teich(S ) 2T Teich(S ), X g X g ∧ → ∧ 1991 Mathematics Subject Classification. 30F60, 53C21, 32G15. e Keywordsandphrases. Teichmu¨llerspace,Weil-Peterssonmetric,Curvatureoperator, Holomorphic sectional curvature. 1 2 MICHAELWOLF&YUNHUIWU anendomorphismofthe(3g 3)(6g 7)-dimensional exterior wedgeproduct − − space 2T Teich(S ) of the tangent space T Teich(S ) at X Teich(S ). X g X g g ∧ ∈ Of course, one popular (and important) way to interpret the “curvature” of a manifold is through its sectional curvatures: while these arise as diagonal elements of the curvature operator Q, our interest in the first part of the paper is on the full operator Q, at least in terms of estimating its spectrum. In [40], by applying the Tromba-We olpert Weil-Petersson curvature formula, one of us characterizedethe zero level set of the curvature operator Q and showed that the Riemannian curvature operator of the Weil-Petersson metricisnon-positivedefinite; inparticular,someoftheeigenvalues ofQaree zero and some are negative. In the first part of this paper, we develop some results for the negative eigenvalues, especially as the genus of the surefaces represented in the moduli space becomes arbitrarily large. We survey these in the next four subsections. In the second part of the paper – described in more detail in the fifth subsection of this introduction – we turn our attention to holomorphic sectional curvatures. We find, for hyperbolic surfaces X of genus g of suffi- g ciently large injectivity radius, that we may exhibit holomorphic sections whose curvatures are uniformly bounded away from zero; here the bound does not depend on the genus of the surface but only on the lower bound we have chosen for the injectivity radius. 1.1. Uniform lower bounds on the norm of the curvature operator. There are recent suggestions that as the genus g of S grows large, some g regions in the Teichmu¨ller space Teich(S ) should become increasingly flat. g We show that, from the point of view of the full curvature operator, this is not true: on any sequence of surfaces with genus tending to infinity, at least one of the eigenvalues of Q does not tend to zero. In particular, let λ (X) min denote the minimal, i.e. the most negative, eigenvalue of Q at X. Our first result is e e Theorem 1.1. For any X Teich(S ) and let Sca(X) be the scalar curva- g ∈ ture at X, then 1 λ (X) 6 − < 0. min 2π More precisely, 2Sca(X) λ (X) 6 . min 9(g 1) − Because this eigenvalue is the maximum of values of Q applied to (unit) elements of 2T Teich(S ), it is enough to estimate Q on a carefully chosen X g ∧ element of 2T Teich(S ) which yields the bound. e X g ∧ e 1.2. Uniform upper bounds on the norm of the curvature operator on the thick locus. It is clear that λ (X) is less than or equal to the min UNIFORM BOUNDS 3 sectional curvatures at X along arbitrary planes. However, it is well-known that the sectional curvatures of Teich(S ) along certain planes could be g arbitrarily negative. One specific example is the direction along which a nontrivial simple closed curve of S is pinching to zero. Then that sectional curvature, along the holomorphic plane defined by the pinching direction, tends to negative infinity as the curve goes to zero. More precisely, we let α beanontrivialsimpleclosedcurveonS andl (X)bethelengthoftheclosed α 1 geodesic in X representing α. Consider the gradient λ := grad(l2) whose α α magnitude approaches 1 as l goes to zero. Wolpert in [39] proved that √4π α the sectional curvatures along holomorphic planes spanned by λ behave as α 3+O(l ), whichgoestonegative infinityasαpinchestozero. Thisrefined π−lα α an earlier estimation of Huang [9]. For more details, one can see Corollary 16 in [39]. Thus, in general we do not have any uniform lower bounds for λ (X) over Teich(S ). min g On the other hand, the Mumford compactness theorem [20] implies that thethickpartofthemodulispaceiscompact. Sincethemappingclassgroup acts on Teich(S ) by isometries, the Weil-Petersson curvature operator Q, g restricted to the thick part of Teich(S ), is bounded. In particular the g minimal eigenvalue λ (X) has a lower bound when X runs over the thicek min part of Teich(S ). Our next result implies that the lower bound may be g taken to be uniform, independent of the topology of the surface. Theorem 1.2. Given an ǫ > 0, let Teich(S )>ǫ be the ǫ-thick part of g Teich(S ). Then, there exists a constant B(ǫ) > 0, depending only on ǫ, g such that λ (X) > B(ǫ), X Teich(S )>ǫ. min g − ∀ ∈ The argument will present an explicit constant B(ǫ) whose asymptotics as ǫ 0, we will exploit in later sections of the paper. In addition, a → ∞ direct corollary is that the sectional curvature of the Weil-Petersson metric, restricted on the thick part of the moduli space, is uniformly bounded from below, a result due originally to Huang [10]. 1.3. Uniform pinched bounds on the norm of the curvature operator on sequences of increasingly thick surfaces. The subject of the asymptoticgeometryofM ,themodulispaceofS ,asgtendstoinfinity,has g g recentlybecomequiteactive: seeforexampleMirzakhani[21,22,23,24]and Cavendish-Parlier[6](resultsobtainedbyrefiningBrock’s [5]quasi-isometry of Teich(S ) to the pants graph). In terms of curvature bounds, by com- g bining the results in Wolpert [34] and Teo [29], we may see that, restricted on the thick part of the moduli space, the scalar curvature is comparable to g as g goes to infinity. In the sixth section of this paper, we study − the asymptotic behaviors of certain eigenvalues of Q as the genus goes to infinity. e 4 MICHAELWOLF&YUNHUIWU We recall for context that Buser and Sarnak [4] proved that there exists a family of closed hyperbolic surfaces X of genus g with g as k k k k and inj(X ) > 2lngk. Now, we noted above that the constan→t B∞(ǫ) in T→he∞o- k 3 rem 1.2 may be explicitly stated. By examining the asymptotic property of that constant B(ǫ) as ǫ , we prove that the minimal eigenvalue of the → ∞ Weil-Petersson curvature operator on a surface of large injectivity radius is pinched by two explicit negative numbers. More precisely, Theorem 1.3. Let X be a sequence of hyperbolic surfaces whose injec- g { } tivity radii satisfy lim inj(X )= . Then, for g large enough, g g →∞ ∞ 25 1 − 6 λ (X ) 6 − . min g π 2π 1.4. Bounds for small negative eigenvalues in the thick part of M g for large genus g. Thus far, we have focused exclusively on the minimal eigenvalueλ (X ). Inthelatter portionofthispartofthepaper,westudy min g some other eigenvalues of the Weil-Petersson curvature operator and their dependence on the genus g. In section 3, we restate Theorem 1.1 in [40] in terms of the eigenvalues of the Weil-Petersson curvature operator. More precisely, Theorem 1.4. For any X Teich(S ), the Weil-Petersson curvature op- g ∈ erator Q at X has exactly (3g 3)2 negative eigenvalues. − Thus, for any X Teich(S ), one may list the set of all non-zero eigen- g e ∈ values of the Weil-Petersson curvature operator at X as follows λ (X) 6 λ (X) 6 λ (X) 6 λ (X) < 0. (3g 3)2 (3g 3)2 1 2 1 − − − ··· Theminimaleigenvalueλ (X) = λ (X)basicallymeasuresthenorm min (3g 3)2 − of the curvature operator Q at X. Fixing i 1,2, ,(3g 3)2 , the i-th eigenvalue λ (X) is a continuous i ∈{ ··· − } function on Teich(Sg) sinceethe Weil-Petersson metric is smooth. Thus, the Mumford compactness theorem implies that the function λ (X) achieves its i minimaand maxima in the thick partTeich(S )>ǫ for fixed ǫ > 0; we denote g those minima and maxima by λǫ(g) and λǫ(g) respectively. In this nota- i i tion, focusing our attention now on the index i, Theorem 1.1 and Theorem 1.2 state that both λǫ (g) and λǫ (g) are pinched by two negative (3g 3)2 (3g 3)2 numbers which are inde−pendent of the−genus of the surface. In section 6 we show that, if that index i is not close to (3g 3)2, then both λǫ(g) and i ǫ − λ (g) could be arbitrarily close to zero, once we take g large enough. More i precisely, Theorem 1.5. If the function f satisfies limsup f(g) = α < 1. Then, g 9g2 →∞ for g sufficiently large, we have −B(α,ǫ) 6 λǫ (g) 6 λǫ (g) < 0 g f(g) f(g) UNIFORM BOUNDS 5 where B(α,ǫ) is a constant only depending on α and ǫ. In particular, lim λǫ(g) = λǫ(g) = 0, for all 16 i 6 8g2. i i g →∞ A direct consequence is Corollary 1.6. Fix ǫ > 0, then for any ǫ > 0, the probability 0 Prob 1 6 i 6 (3g 3)2; λǫ0(g) < ǫ 1 as g . { − | i | } → → ∞ Proof. Let p(g) := Prob 1 6 i 6 (3g 3)2; λǫ0(g) < ǫ . Theorem 1.5 { − | i | } then says that, for any 06 α < 1, α 6 liminfp(g) 6 limsupp(g) 6 1. g g →∞ →∞ Since α is arbitrary in [0,1), lim p(g) = 1. (cid:3) g →∞ Mirzakhani (Theorem 4.2 in [24]) proved that for a small enough number ǫ > 0, the volume of the ǫ -thin part of the moduli space M is comparable 0 0 g to ǫ2Vol(M ) as g . Corollary 1.6 suggests that the moduli space M 0 g → ∞ g tends to be flat as g goes to infinity in a probabilistic sense. 1.5. Existence of holomorphic lines on sufficiently thick surfaces with uniformly pinched Weil-Petersson holomorphic sectional curvatures. In [30, 34] it was shown that the sectional curvature of the Weil- Petersson metric for Teich(S ) is negative and the holomorphic sectional g curvature is bounded above by a negative number comparable to 1. The −g dependence of this bound on g begs the question as to whether there are bounds on the sectional curvature that are independent of the topology of the surface, even if one allows an additional restriction to a thick part of modulispace. However, Teo in [29] showed that, restricted to any thick part of the moduli space, the Ricci curvature is uniformly bounded from below. As the Ricci curvature is a trace over 6g 7 curvatures, we then see that − some sectional curvatures go to zero along any sequence of surfaces X in g the thick part of Teich(S ) as g . This aslo suggests the question1 g → ∞ whether, Question 1. Restricted to any thick parts of the moduli spaces, do all of the sectional curvatures tend to zero as the genus goes to infinity? The goal of the second part of this paper is to show, for the portion of theTeichmu¨ller spacecomposedofsurfaceswhichare“sufficiently thick”–a termwewilldefinepreciselyinDefinition2.14–theexistenceofholomorphic lines whose holomorphic sectional curvatures are uniformly bounded away from zero. This gives a negative answer to the question above. In notational preparation for themore precise version of the result, we re- callthataholomorphicsectionofthe(complexified)tangentspaceT Teich(S ) X g 1We are told by Zheng Huang that Question 1 is originally raised by Maryam Mirza- khani. Wewould like to thankboth of them here. 6 MICHAELWOLF&YUNHUIWU ofX Teich(S )intheTeichmu¨llerspaceTeich(S )isdefinedbyaharmonic g g ∈ Beltrami differential µ. Theorem 1.7. There is a constant ι so that if X Teich(S ) is suffi- 0 g ∈ ciently thick, i.e. inj(X) > ι , then there exists a µ in the tangent space 0 X T Teich(S ) so that the Weil-Petersson holomorphic sectional curvature X g K(µ ) along µ satisfies X X 81C K(µ ) 6 − 0 < 0 X 6400 π2 · where C = 2 . 0 3C(1)2 Vol (B(0;1)) · D The expressions C(1) refers to an explicit function C(inj(X)) of the injectivity radius inj(X) which we shall display in Definition 2.8. In addition, thereis also a uniformlower boundfor K(µ ) which is dueto Huang in [10] X (see also Theorem 1.2). Analmostimmediatecorollaryofourmethodofproofreferstoasequence of surfaces X of growing genus whose injectivity radii inj(X ) as g g → ∞ g . → ∞ Corollary 1.8. Let X be a sequence of hyperbolic surfaces whose in- g { } jectivity radii satisfy lim inj(X ) = . Then, there exists a uniform g g →∞ ∞ constant E > 0 such that for g large enough, the Weil-Petersson holomorphic sectional curvatures satisfy 2 6 min K(σ ) 6 E < 0 g −π σg⊂TXgTeich(Sg) − where the minimum runs over all the holomorphic lines in T Teich(S ). Xg g [We note that the above Theorem 1.7 also proves that the minimal (most negative) eigenvalue of the Weil-Petersson curvature operator, restricted on sufficiently thick hyperbolic surface, is uniformly bounded away from zero. Inthatsense,thisresultthenmakescontact with,andiscertainly consistent with,Theorem1.1,butthetworesultsaredistinct,sinceourexamplehereis restrictedtoholdonthisspecial‘sufficientlythick’region(and,furthermore, the asserted constants are different).] 1.6. A remark on choice of normalizations. Some of the uniformity of the bounds that are independent of genus of course results from the conventions of fixing the uniformized metrics on the Riemann surfaces to all have curvature identically 1 (instead of, say, curvature identically g), − − forcing the areas of the surfaces to grow linearly with g in area. Other conventionsonrepresentativesofthemodulispacewouldtranslateintoother bounds, but the ones we choose are the ones that seem most prevalent in the present literature. UNIFORM BOUNDS 7 1.7. Discussion of the methods. The methods for the two parts of the paper are distinct. In the first part, we begin with the formula of the Weil- Petersson curvatureoperatorQin[40]: seeequation (3.4). Thatformulaex- hibits< Q(A),A >asasumoftermsinvolvingtheoperator 2(∆ 2) 1 and − − − algebraic expressions in a holoemorphic orthonormal basis of T Teich(S ). X g By applyeing some fundamental properties of the operator 2(∆ 2) 1, we − − − give lower and upper bounds for < Q(A),A > in terms of, for our appli- cations, quantities that only involve pointwise values of the holomorphic basis. Here we briefly outline our proeofs of Theorem 1.1, Theorem 1.2 and Theorem 1.7. Fortheproofof Theorem 1.1: Byapplyingourupperboundforthecur- vatureoperatorformulatothemostsymmetricelementA 2T Teich(S ) X g ∈ ∧ we show that (up to a positive constant) < Q(A),A > is bounded above by the scalar curvature of the Weil-Petersson metric at X, scaled by a factor of 1. Then we apply Wolpert’s upper boundeof the scalar curvature which g involves a factor of g to finish the proof. For the proof of Theorem 1.2: Choose an arbitrary element B of unit length in 2T Teich(S ). Our lower bound of < Q(B),B > involves only X g ∧ terms that are products of our holomorphic basis, weighted by the coeffi- cientsoftheelementB inthatbasis. Well-known boeundsontermslikethese then yield a bound in terms of those coefficients and an explicit function of the injectivity radius. The condition that B is of unit norm then yields an expression that is independent of genus. For the proofof Theorem 1.7: We choose a holomorphicline that corre- spondsto theimage of a constant function on thediskundertheΘ-operator to the hyperbolic surface X , here thought of as the quotient of H2 under g the action of a Fuchsian group Γ . The resulting Poincaré series has a term g corresponding to the identity element of Γ and a series of other ‘error’ g terms. We adapt a method of Ahlfors [2] to show that outside a large ball, the contribution of the error terms is bounded by a subharmonic function. Taking the ball large enough, we can show that the contribution of the error terms, now estimated by its value on the boundary of the large ball – in its role as the boundary of the region of subharmonicity – can be made arbitrarily small. Thus only the term corresponding to the identity element of Γ cannot be made arbitrarily small, and estimates of that term yield the g desired bound. 1.8. Plan of the paper. Section 2 provides some necessary background andthebasicpropertiesoftheWeil-Petersson geometryofTeichmu¨llerspace that we will need. After that preparatory section, the paper splits into two parts, each of which may be read independently of the other. The first part treats our results on the Weil-Petersson curvature operator, while the second part discusses the curvature of some holomorphic sections of the tangent bundle over very thick parts of Teichmu¨ller space. 8 MICHAELWOLF&YUNHUIWU Part I begins with section 3 in which weestimate theformula of theWeil- Petersson curvature operator and restate the non-positivity of the Weil- Petersson curvature operator in terms of eigenvalues. Section 4 provides the upper bound Theorem 1.1 for the minimal eigenvalue. In section 5 we establish Theorem 1.2, the lower bound for the minimal eigenvalue in the thick part. We prove Theorem 1.3 and Theorem 1.5 in section 6. Part II spans two sections. We begin in section 7 with some estimates on harmonic Beltrami differentials, and then apply those estimates in section 8 to prove Theorem 1.7 and Corollary 1.8. Acknowledgement. The authors would like to thank Maryam Mirzakhani, Hugo Parlier and Kasra Rafi for useful conversations. They also would like to thank Scott Wolpert for useful conversations and suggestions on part (2) of this article. They deeply thank Zheng (Zeno) Huang for bringing Question 1 to their attention, his detailed comments for this article, and for his support over many years. The first author also gratefully acknowledges support from the U.S. National Science Foundation through grant DMS 1007383. This work was partially completed while the second author visited the Math- ematical Sciences Center of Tsinghua university. Both authors acknowl- edge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric structures And Representation vari- eties”(the GEAR Network). 2. Notations and Preliminaries 2.1. Notation and Background on Weil-Petersson curvatures. In this section, we set our notations and quickly review the relevant background material on the Weil-Petersson metric and curvatures. We have two principal goals. For use in Part I, we prove Proposition 2.5, which shows that the scalar curvatures are comparable to the L2 norms of a pointwise Bergman sum of harmonic Beltrami differentials. For use in Part II, we prove Proposition 2.11, which shows that the holomorphic sectional curva- turealongaharmonicBeltramidifferential µisestimated intermsofpowers of its normalized L norm. ∞ To begin, recall that we denoted by S a closed oriented surface of genus g g > 2. We may equip S with a hyperbolic metric σ(z)dz 2, here written g | | in a local conformal coordinate z induced by the metric. Again, Teich(S ) g is the Teichmu¨ller space of surfaces of genus g, which we may construe as equivalence classes under the action of the group Diff of diffeomorphisms 0 isotopic to the identity of the space of hyperbolic surfaces X = (S ,σ dz 2). g | | The tangent space T Teich(S ) at a point X = (S ,σ dz 2) is identified X g g | | with the space of harmonic Beltrami differentials on X, i.e. forms on X expressible as µ = ψ where ψ H0(X,K2) is a holomorphic quadratic σ ∈ differential on X. Let z = x+iy and dA = σ(z)dxdy be the volume form. The Weil-Petersson metric is the Hermitian metric on Teich(S ) arising g UNIFORM BOUNDS 9 from the the Petersson scalar product ϕ ψ < ϕ,ψ >= · dA σ2 ZS via duality. We will concern ourselves primarily with its Riemannian part g . ThroughoutthispaperwedenotetheTeichmu¨ller spaceendowed with WP the Weil-Petersson metric by Teich(S ). g Set D = 2(∆ 2) 1 where ∆ is the Beltrami-Laplace operator on − − − X = (S,σ dz 2). The operator D is positive and self-adjoint. The following | | inequality follows from the maximum principle; see Lemma 5.1 in [32] for details. Lemma 2.1 ([32]). For any harmonic Beltrami differential µ on X, we have µ 2 D(µ 2)> | | . | | 3 Thefollowingpropertyiswell-knowntoexperts–seeforexampleLemma4.3 in [16]. For completeness, we include the proof. Lemma 2.2. Let D be the operator above. Then, for any complex-valued function f C (X), ∞ ∈ 0 6 (D(f)f)dA 6 f 2dA. | | ZX ZX Proof. Let g = D(f), so that f = 1(∆ 2)g. The left inequality follows −2 − directly by integrating by parts. For the right inequality, begin by assuming that f is real-valued. In that casedecomposef = ∞i=0φi asalinearcombination ofeigenfunctions ofthe Laplacian: here φ satisfies (φ φ )dA = 0 for all i = j and ∆φ = µ φ , i P X i j 6 i i i where µ < 0 is the i-th eigenvalue of the Beltrami-Laplace operator ∆. i R Since (φ φ )dA = 0 for all i= j, we have X i j 6 R ∞ 2φi ∞ (D(f)f)dA = ( )( φ )dA i 2 µ ZX ZX i=0 − i i=0 X X ∞ 2 = φ 2dA i 2 µ | | ZX i=0 − i X ∞ 6 φ 2dA i | | ZX i=0 X = f2dA. ZX If f is complex-valued, one applies the same method to the real and imagi- nary parts of f separately, along with a standard use of the self-adjointness of D. (cid:3) 10 MICHAELWOLF&YUNHUIWU WeconcludethisdiscussionoftheoperatorD = 2(∆ 2) 1 byrecording − − − some elementary properties of its Green’s function. Proposition 2.3. Let D be the operator D = 2(∆ 2) 1. Then there − − − exists a Green function G(w,z) for D satisfying: (1). D(f)(z) = G(z,w)f(w)dA(w) for any f C ((X,σ dz 2),C). w X ∈ ∞ | | (2). G(w,z) is pos∈itive. R (3). G(w,z) is symmetric, i.e, G(w,z) = G(z,w). Proof. See for example [26] and [34]. (cid:3) 2.2. The Riemannian tensor of the Weil-Petersson metric. Thecur- vature tensor of the Weil-Petersson metric is given as follows. As described in the opening paragraph of section 2.1, let µ ,µ be two elements in the i j tangentspaceT Teich(S )atX, sothatthemetrictensor mightbewritten X g in local coordinates as g = µ µ dA. ij i· j ZX For the inverse of (g ), we use the convention ij gijg = δ . kj ik Then the curvature tensor is given by ∂2 ∂ ∂ R = g gst g g . ijkl ∂tk∂tl ij − ∂tk it∂tl sj The following curvature formula was established in [30, 34]. One can also see [13] for a derivation from a third perspective. It has been applied to study various curvature properties of the Weil-Petersson metric. In [27] Schumacher showed that Teich(S ) has strongly negative curvature in the g sense of Siu. Huang in his thesis [9] showed that the sectional curvatures of Teich(S ) can not be bounded away from zero. Liu-Sun-Yau in [15] showed g that Teich(S ) has dual Nakano negative curvature, which says that the g complex curvature operator on the dual tangent bundle is positive in some sense. Motivated by the method in [15], the second author in [40] showed that the Teich(S ) has negative semi-definite Riemannian curvature opera- g tor. Onecanalsosee[16,17,18,29,37,39]forotheraspectsofthecurvature of Teich(S ). g Theorem 2.4 (Tromba, Wolpert). The curvature tensor satisfies R = D(µ µ ) (µ µ )dA+ D(µ µ ) (µ µ )dA. ijkl i j · k l i l · k j ZX ZX