arXiv:1205.5878v2 [math.DG] 28 Oct 2016 et ubrb number Betti uvtrslei ( in homeomorphic lie is curvatures Riemannian compact connected ple T n hoe 1.3. Theorem 2.7) theorem [2] eodBtinme b number Betti second oe o ayeape.I hsnt esalueRcicurva Ricci use shall we note theore this that know In we su theorems examples. two many Poinca the too of From cover help the sphere. with a manifo Hence to Riemannian morphic sphere. connected homotopy a simply is compact curvature a that says [5]) aueRic vature hoe 1.1. Theorem [6]) pla (c.f. It result classical number technique. Bochner’s Betti Bochner and is curvature between tool relation strong understanding the of One geometry. ups htni d n h etoa uvtr aifisth satisfies curvature sectional the and odd is n that Suppose hoe 1.2. 2.8) Theorem theorem [2] see also [1], (c.f. following p ffi ≥ EUTO IC UVTR N H EODBTINUMBER BETTI SECOND THE AND CURVATURE RICCI ON RESULT A M ealta h ac-egrKignegsshr theore sphere Rauch-Berger-Klingenberg’s the that Recall eepstv stoi uvtr en,frayfu othon four any for means, curvature isotropic positive Here osdradi a Consider h eerhi upre yteNtoa aua cec Fo Science Natural National the by supported is research The phrases. and words Key 2010 h td frlto ewe uvtr n oooyi h c the is and curvature between relation of study The egrivsiae hti htcs h eodBtinumbe Betti second the case what in that investigated Berger in odto o h eodBtinme aihn.Orm Our vanishing. number Betti second the for condition cient 4 h uvtr esrsatisfies tensor curvature the , ups htni vnadMhspstv stoi curvatur isotropic positive has M and even is n that Suppose . ahmtc ujc Classification. Subject Mathematics A ndim in K¨ahler-Einstein manifold compact conseq for As restriction restriction. curved curved Ricci certain under vanishes M bstract > = 2 0 Bcnr14)LtMb opc imninmnfl ihRi with manifold Riemannian compact a be M Let 1946) (Bochner Mcle-ag e eacmatReana aiodo di of manifold Riemannian compact a be M Let (Micallef-Wang) Bre)LtMb opc imninmnfl fdimension of manifold Riemannian compact a be M Let (Berger) ( hntefis et ubrb number Betti first the Then . epoeta h eodBtinme facmatRiemannia compact a of number Betti second the that prove We . 4 ff M , rn uvtr odto,Mcle n agpoe cf [ (c.f. proved Wang and Micallef condition, curvature erent 1 4 5. ) , = ] eeaiaino peeterm(ust Micallef-Mo to (dues theorem sphere of generalization A 1]. 2 0 ic uvtr,Btinumber. Betti curvature, Ricci ( . M R ) 1313 = 0 + . R 1414 1. rmr 32;Scnay53C25. Secondary 53C20; Primary INIGWAN JIANMING + introduction R 2323 1 1 + ( M R 2424 ) = naino hn N0.11301416. China of undation n oooyshr theorem sphere a and s 0 > . ecsw bana interesting an obtain we uences 2 | R .Tefis euti hsfil is field this in result first The s. 1234 cf 1)sae htasim- a that states [1]) (c.f. m at ra vectors ormal savr motn oein role important very a ys nrltpci Riemannian in topic entral | oashr ftesectional the if sphere a to ecnetr ti homeo- is it conjecture re . 4 aihs epoe the proved He vanishes. r dwt oiieisotropic positive with ld n n − i eutis result ain − uet iearelaxedly a give to ture 3 s12ad13cnnot can 1.3 and 1.2 ms 9 ≤ .Te h second the Then e. K manifold n M < e 1 1 , ] losee also 4], e hnthe Then . 2 , mension c cur- cci e r c.f. ore n 3 , ≥ e 4 5 ∈ . 2 JIANMING WAN

Theorem 1.4. Let M be a compact Riemannian manifold. The dimension dim(M) = 2m or 2m + 1. Let k¯ (resp. k) be the maximal (resp. minimal) sectional curvature of M. If the Ricci curvature of M satisfies that 2m − 2 (1.1) Ric > k¯ + (k¯ − k), M 3 then the second b2(M) = 0. Particularly, if M is a compact Riemannian manifold with nonnegative sectional curva- ture, then the second Betti number vanishes provided 2m + 1 (1.2) Ric > k¯. M 3 Note that there is no dimensional restriction in theorem 1.4. Any compact K¨ahler manifold does not satisfy (1.1) since it has b2 ≥ 1. The condition 1.1 is a Ricci pinching condition. We mention that several other Ricci pinching type theorems obtained by Gu and Xu (c.f. [3] [7], ). As an immediate consequence, we obtain a curvature restriction for special Einstein . Corollary 1.5. Let M be a compact Einstein manifold with nonzero second Betti number. Then the Ricci curvature satisfies 2m − 2 (1.3) Ric ≤ k¯ + (k¯ − k). 3 In addition, if the sectional curvature is nonnegative, one must have 2m + 1 (1.4) Ric ≤ k¯. 3 Particularly (1.3) holds for any compact K¨ahler-Einstein manifold. Remark 1.6. 1) The condition (1.1) implies that the maximal sectional curvature k¯ > 0: If k¯ ≤ 0, then 2m − 2 k¯ ≥ Ric > k¯ + (k¯ − k). M 3 We get k¯ < k. This is a contradiction. 2) Since k¯ > 0, of course (1.1) implies RicM > 0. 3) If the minimal sectional curvature k < 0. Since k¯ > 0. If dim(M) = 2m + 1, from 2m − 2 2mk¯ ≥ Ric > k¯ + (k¯ − k), M 3 one has 2m − 2 k¯ > |k|. 4m − 1 Similarly 1 k¯ > |k| 2 provided dim(M) = 2m. We use theorem 1.4 to test some simple examples. ARESULTONRICCICURVATUREANDTHESECONDBETTINUMBER 3

Example 1.7. 1) The space form S n, k¯ = k = 1, Ric = n − 1 = k¯ for n = 2 and Ric = 2 n n − 1 > k¯ for n , 2, b2(S ) = 1 and b2(S ) = 0 for n , 2. 2 2 ¯ = = = ¯ + 2n−2 ¯ 2 2 = 2) S × S with product metric, k 1, k 0, Ric 1 < k 3 (k − k), b2(S × S ) 2. m m ¯ = = = 2m+1 ¯ = 3) S × S , m > 4 with product metric, k 1, k 0, Ric m − 1 > 3 k, b2 0. CPn ¯ = = = + = ¯ + 2n−2 ¯ 4) with Fubini-Study metric, k 4, k 1, Ric 2n 2 k 3 (k − k), n b2(CP ) = 1. From the examples we know that the inequality (1.1) is sharp. The proof of theorem 1.4 is also based on Bochner technique. But comparing with Berger and Micallef-Wang’s results, we consider a different side. This allows us get a uniform result (without dimensional restriction).

2. Proof of the theorem 2.1. Bochner formula. Let M be a compact Riemannian manifold. Let ∆= dδ + δd be the Hodge-Laplacian, where d is the exterior differentiation and δ is the adjoint to d. Let ϕ ∈ Ωk(M) be a smooth k-form. Then we have the well-known Weitzenb¨ock for- mula (c.f. [6]) (2.1) ∆ϕ = ∇2 ϕ − ωi ∧ i(v )R ϕ, vivi j viv j Xi Xi, j 2 = = + + here ∇XY ∇X∇Y − ∇∇X Y and RXY −∇X∇Y ∇Y ∇X ∇[X,Y]. The {vi, 1 ≤ i ≤ n} are the local orthonormal vector fields and {ωi, 1 ≤ i ≤ n} are the duality. A k-form ϕ is called harmonic if ∆ϕ = 0. k The famous Hodge theorem states that the HdR(M) is isomorphic to the space spanned by k-harmonic forms. = i j Let ϕ i, j ϕi jω ∧ ω be a harmonic 2-form. By (2.1), under the normal frame we can get (c.f. [2]P or [1])

(2.2) ∆ϕi j = (Ricikϕk j + Ric jkϕik) − 2 Rik jlϕkl, Xk Xk,l = = where Ri jkl hR(vi, v j)vk, vli is the curvature tensor and Rici j khR(vk, vi)vk, v ji is the Ricci tensor. P So we have 2 2 ∆|ϕ| = 2 ϕi j∆ϕi j + 2 (vkϕi j) Xi, j Xi, j Xk

≥ 2 ϕi j∆ϕi j Xi, j , 2F(ϕ).

Note that by (2.1) one has the global form of above formula 1 0 = −h∆ϕ, ϕi = |∇ ϕ|2 + h ωi ∧ i(v )R ϕ, ϕi− ∆|ϕ|2. vi j viv j 2 Xi Xi, j

i The F(ϕ) is just the curvature term h i, j ω ∧ i(v j)Rviv j ϕ, ϕi. P 4 JIANMING WAN

2.2. Proof of Theorem 1.4. By Hodge theorem, we only need to show that every har- monic 2-form vanishes. Case 1: Assume dim(M) = 2m. For any p ∈ M, we can choose an orthonormal basis = ∗ ∗ {v1, w1,..., vm, wm} of T p M such that ϕ(p) α λαvα ∧ wα (for instance c.f. [1] or [2]). ∗ ∗ Here {vα, wα} is the dual basis. Then P m m = 2 + (2.3) F(ϕ) λα[Ric(vα, vα) Ric(wα, wα)] − 2 λαλβR(vα, wα, vβ, wβ) Xα=1 α,βX=1 The term m m = 2 −2 λαλβR(vα, wα, vβ, wβ) −2 λα · λβ · R(vα, wα, vβ, wβ) − 2 λαR(vα, wα, vα, wα) α,βX=1 Xα,β Xα=1 4 m ≥ − (k¯ − k) |λ |·|λ |− 2k¯ λ2 3 α β α Xα,β Xα=1 2 ≥ − (k¯ − k) (λ2 + λ2) − 2k¯|ϕ|2 3 α β Xα,β 2 = − (k¯ − k)(2m − 2)|ϕ|2 − 2k¯|ϕ|2 3 2m − 2 = −2[k¯ + (k¯ − k)]|ϕ|2. 3 The first ” ≥ ” follows from Berger’s inequality (c.f. [1]): For any orthonormal 4-frames {e1, e2, e3, e4}, one has 2 |R(e , e , e , e )|≤ (k¯ − k). 1 2 3 4 3 On the other hand, by the condition (1.1) we have m 2m − 2 λ2 [Ric(v , v ) + Ric(w , w )] ≥ 2[k¯ + (k¯ − k)]|ϕ|2, α α α α α 3 Xα=1 the equality holds if and only if ϕ(p) = 0. This leads to F(ϕ) ≥ 0 with equality if and only if ϕ(p) = 0. Since 1 F(ϕ) ≤ ∆|ϕ|2 = 0, ZM 4 ZM we get F(ϕ) ≡ 0. Thus the harmonic 2-form ϕ ≡ 0. Case 2: If dim(M) = 2m + 1. For any p ∈ M, we also can choose an orthonormal basis = ∗ ∗ {u, v1, w1,..., vm, wm} of T p M such that ϕ(p) α λαvα ∧ wα (c.f. [1] or [2]). We also have m P m = 2 + F(ϕ) λα[Ric(vα, vα) Ric(wα, wα)] − 2 λαλβR(vα, wα, vβ, wβ). Xα=1 α,βX=1 Thus the argument is same to the even dimensional case. This completes the proof of the theorem. ARESULTONRICCICURVATUREANDTHESECONDBETTINUMBER 5

3. Sphere theorem in dim4 and 5 Theorem 3.1. Let M be a compact Riemannian manifold. dim M = 4 or 5. If 5k¯ − 2k Ric > , M 3 then M is a real homology sphere, i.e. bi(M) = 0 for 1 ≤ i ≤ dim M − 1.

Proof. Since RicM > 0, from theorem 1.1 we know that b1(M) = 0. Theorem 1.4 implies that b2(M) = 0. With the help of Poincare duality, we obtain the theorem.  Finally we metion a differential sphere theorem for Ricci curvature obtained by Gu and Xu ( c.f. [3] theorem D). Theorem 3.2. Let M be a simple conncted compact Riemannian n-manifold. If 11 Ric > (n − )k¯, M 5 then M is diffeomorphic to S n.

References 1. M. Berger, Sur quelques vari´et´es riemaniennes suffisamment pinc´ees, Bull. Soc. Math. France 88, (1960) 57-71. 2. S. Brendle and R. Schoen Sphere theorems in geometry, Surveys in differential geometry. Vol. XIII. Geome- try, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, 49-84, Surv. Differ. Geom., 13, Int. Press, Somerville, MA, 2009. 3. J.GuandH. Xu, The sphere theorems for manifolds with positive scalar curvature. J. Differential Geom. 92 no 3, 507-545, 2012. arXiv:1102.2424v1 4. M. Micallef and M. Wang, Metrics with nonnegative isotropic curvature, Duke Math. J. 72, no. 3, (1993) 649-672. 5. M. Micallef and J.D. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. 127, 1988, 199-227. 6. H. Wu, The Bochner technique in differential geometry. Math. Rep. 3 (1988), no. 2, i-xii and 289-538. 7. H. Xuand J. Gu, The differentiable sphere theorem for manifolds with positive Ricci curvature. Proc. Amer. Math. Soc. 140 (2012), no. 3, 1011-1021.

Department of Mathematics,Northwest University,Xi’an 710127, China E-mail address: wanj [email protected]; [email protected]