WEYL STRUCTURES WITH POSITIVE RICCI TENSOR

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1 Available at: IC/99/156 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS WEYL STRUCTURES WITH POSITIVE RICCI TENSOR B. Alexandrov 1 Faculty of Mathematics and Informatics, Department of Geometry, University of Sofia, 5 James Bourchier Blvd, 1126 Sofia, Bulgaria and S.Ivanov 2 Faculty of Mathematics and Informatics, Department of Geometry, University of Sofia, 5 James Bourchier Blvd, 1126 Sofia, Bulgaria and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract We prove the vanishing of the first Betti number on compact manifolds admitting a Weyl structure whose Ricci tensor satisfies certain positivity condition, thus obtaining a Bochner-type vanishing theorem in Weyl geometry. We also study compact Hermitian-Weyl manifolds with non-negative symmetric part of the Ricci tensor of the canonical Weyl connection and show that any such manifold has first Betti number b1 = 1 and Hodge numbers h p, = 0 for p > 0, h 0,1 = 1, h,q = 0 for q > 1. MIRAMARE - TRIESTE October alexandrovbt@fmi.uni-sofia.bg ivanovsp@fmi.uni-sofia.bg

2 1 Introduction A Weyl structure on a conformal manifold (M, c) is a torsion-free linear connection V w preserving the conformal structure c. This implies that for any Riemannian metric g G c there exists a 1-form θ such that V w g = θ <S> g. Conversely, given a 1-form θ and g G c, there exists a unique Weyl structure V w such that V^g = θ g. Weyl structures arise naturally for example in almost Hermitian geometry (the Lee form determines the Weyl connection) and contact geometry (the contact form determines the Weyl connection). More generally, when the geometry has a preferred 1-form, then the Weyl structure is determined by the given metric and this 1-form. The Weyl structure V w is called closed (resp. exact) if θ is closed (resp. exact). The well known theorem of Gauduchon [?] says that if M is compact and at least 3-dimensional, then there exists a unique (up to homothety) metric gee, such that the corresponding 1-form 6 is co-closed with respect to g. Recently there has been a considerable interest in Weyl geometry, mainly in Einstein-Weyl manifolds. A Weyl structure is called Einstein-Weyl if the symmetric trace-free part of its Ricci tensor vanishes. The Einstein-Weyl geometry initially attracted a particular interest in dimension 3 [?,?], but subsequently Einstein-Weyl manifolds have been much studied in all dimensions [?,?,?] (for a nice overview on Einstein-Weyl manifolds see [?]). In the present paper we study Weyl structures on compact manifolds, such that the symmetric part of the Ricci tensor satisfies certain positivity condition. In [?] Gauduchon has shown that a 4-dimensional compact conformal manifold, which admits a closed self-dual Weyl structure with 2-positive normalized Ricci tensor, is conformally equivalent to S 4 or CP 2 with their standard conformal structures. Notice that the 2-positivity condition of Gauduchon implies the positivity of the Ricci tensor of the Weyl connection but the converse is not true. In [?] Pedersen and Swann have proved that on a non-exact compact Einstein-Weyl manifold with vanishing (resp. non-negative but non-identically zero) symmetric part of the Ricci tensor the first Betti number is b1 = 1 (resp. b1 =0). Recently the authors have shown [?] that on a compact Hermitian surface with non-negative everywhere and positive at some point symmetric part of the Ricci tensor of the canonical Weyl connection the first Betti number is b1 = 0. In the present paper we show that b1 = 0 for any compact 3 or 4-dimensional conformal manifold, which admits a Weyl structure with the symmetric part of the Ricci tensor nonnegative everywhere and positive at some point, thus obtaining a vanishing theorem of Bochner type in Weyl geometry: Theorem 1.1 Let (M,c) be a compact conformal manifold of dimension 3 or 4. Suppose that (M, c) admits a non-exact Weyl connection V w such that the symmetric part Ric W of the Ricci tensor ofv w is non-negative on M. Then the first Betti number of M satisfies h < 1.

3 Further, a) If Ric W is strictly positive at some point, then b1 = 0. b) If b1 = 1, then the 1-form θ, corresponding to the Gauduchon metric g of (M,c,V w ), is parallel with respect to the Levi-Civita connection of g and the universal cover of (M,g) is isometric to R x N, where N is of positive Ricci curvature and therefore diffeomorphic to the sphere S 2 or S 3. In particular, if M is an oriented 3-dimensional manifold then M is diffeomorphic but not isometric to S 1 x S 2. Theorem?? is a direct consequence of a more general result (Theorem??), where we prove a vanishing theorem of Bochner type for higher dimensional manifolds supplied with a Weyl connection, whose Ricci curvature satisfies certain positivity condition. Turning to Hermitian manifolds, we recall that the canonical Weyl structure on a Hermitian manifold is the Weyl structure determined by the metric and the Lee form and its definition does not depend on the choice of a metric in the given conformal class. The canonical Weyl structure is called Hermitian-Weyl if it preserves the complex structure (see [?]). This condition is always satisfied in the 4-dimensional case, but for higher dimensions it forces the manifold to be locally conformally Kahler. Every locally conformally Kahler manifold has closed Lee form and a particular subclass of such manifolds is formed by the generalized Hopf manifolds, the Hermitian manifolds whose Lee form is parallel with respect to the Levi-Civita connection. Compact Hermitian Einstein-Weyl manifolds are classified in [?] in dimension 4 and in [?] in higher dimensions. We apply Theorem?? to Hermitian-Weyl 4-dimensional manifolds to get Corollary 1.2 Let (M,J) be a compact complex surface. Suppose that there exists a conformal class c of Hermitian metrics, such that the canonical Weyl structure corresponding to c is nonexact and has Ricci tensor with non-negative symmetric part. Then b1 < 1. Further, (i) Ifb1 = 0, then (M,J) is a rational surface with c1 2 > 0. (ii) Ifb1 = 1, then (M, J) is a Hopf surface. We note that if the symmetric part of the Ricci tensor of the canonical Weyl structure is nonnegative everywhere and positive at some point, then the surface is rational with c 2 > 0 [?]. In dimensions greater than 4 the Hermitian manifolds with non-exact Hermitian-Weyl structures whose Ricci tensor has non-negative symmetric part are generalized Hopf manifolds with respect to the Gauduchon metric according to the results in [?]. It is well known that the first Betti number of a generalized Hopf manifold is odd and therefore these manifolds do not admit any Kahler structure. In general, the first Betti number and the Hodge numbers of a generic generalized Hopf manifold are not known (for surveys on generalized Hopf manifolds see [?,?]). We prove 3

4 Theorem 1.3 Let (M,g,J) be a 2m-dimensional compact generalized Hopf manifold and the canonical Hermitian-Weyl structure has Ricci tensor with non-negative symmetric part. Then b1 = 1 and the Hodge numbers h p,q satisfy the equalities h p, = 0, p = 1,2,...,m, h,q = 0, q = 2,3,...,m, h,1 = 1. 2 Ricci-positive Weyl structures in dimension n In the following we denote by <.,. > and. the pointwise inner products and norms and by (.,.) and. - the global ones respectively. For the curvature and the Ricci tensor of a linear connection V we adopt the following convention: R(X,Y) = [Vx,Vy] - V [X)Y], ρ(x,y)=trace{z > R(Z,X)Y}. Let V w be a Weyl structure on a n-dimensional conformal manifold (M,c). Let j e c and V w g = θ g. The connection V w is given explicitly by (2.1) Vf Y = V X Y - 1 θ(x)y - 1 θ(y)x + 1g(X, Y)θ#, Zi Zi Zi where V is the Levi-Civita connection of g and θ# is the vector field dual to θ with respect to g. We shall denote by Ric W the symmetric part of the Ricci tensor of V w and by k - the conformal scalar curvature, i.e. the trace of Ric W (??) that the Ricci tensor of V w is Ric W + f dθ, with respect to g. It is easy to obtain from (2.2) Ric W (X,X) = ρ(x,x) + 7^(V X 9)(X) - ^( ^ 2 X 2 - θ(x) 2 ) - \\X\ 2 d*9, (2.3) k = s - ( n { n 1 f 2 ) where ρ and s are the Ricci tensor and the scalar curvature of the Levi-Civita connection of g and the norms and the codifferential operator are taken with respect to g. Theorem 2.1 Let (M,c) be a compact n-dimensional (n > 2) conformal manifold, V w be a non-exact Weyl connection on (M,c) and g G c be the Gauduchon metric ofv w. Let θ be the 1-form determined by V w and g and θ# be the vector field dual to θ with respect to g. Suppose that for all tangent vectors X the symmetric part Ric W of the Ricci tensor of V w satisfies (2.4) Ric W (X, X)> {U ~ 2)(n 8 ~ 4) ( θ 2 X 2 - θ(x)2). Then the first Betti number of M satisfies b1 < 1. Further, a) If at some point the inequality (??) is strict for X = θ# (i.e. Ric W (θ#,θ#) > 0), then h = 0. b)ifb1 = 1, then θ is parallel with respect to the Levi-Civita connection of g and the universal cover of (M, g) is isometric to R x N, where the metric on N is of positive Ricci curvature. In particular, for n = 3 and n = 4 the manifold N is diffeomorphic to S n ~ l.

5 Proof: Let V be the connection defined by (2.5) V X Y = V X Y - r^{6{x)y - g(x, Y)θ#). Let <p be a 1-form and ξ be the dual vector field. Then we have the Weitzenbock formula (see e.g. [?], chapter 1) (2.6) It follows from (??) that M (2.7) Since by (??), (??) and the Weitzenbock formula (??) we obtain M / ( M - A \i\ 2 d*9dv. 4 M But g is the Gauduchon metric and thus d*9 = 0. Hence, 2 (2.8) # + IMVII 2 = IIVCII 2 + / (Ric w (U) - { n 2 )^ A \\0\ 2 \i\ 2-0(O 2 )) dv. M 8 Now, let if = 0 be a harmonic 1-form. Then (??) shows that V = 0. Hence Thus, 0 = dtp(x 1,X 2 ) = - 6(X 2 )<p(x{ This shows that θ A if = 0, i.e. θ = ftp for some function f. Hence, ^ = 0, i.e. <p is parallel with respect to the Levi-Civita connection of g. Thus, the universal cover of (M,g) is isometric torxiv (see e.g. chapter 4 in [?]). We can lift θ, if, ξ and f to R x N and we can assume that \<p\ = 1, i.e. ξ is the unit vector field tangent to R. We have (2.9) (V x 0)(X)=<p(X)Xf and since d*9 = d*(p = 0, we obtain

6 It follows from (??), (??) and (??) that (2.11) ρ(x,x) +?Lll ^ ^ Any vector tangent torxjv has the form X = λξ + X- 1, where X 1 - is orthogonal to ξ, and since the metric on R x N is the product metric, we have ρ(x,x) = p(x J -,X- L ). Thus, using (??) and (??) we obtain (2.12) P(X\X ^ ) +n for any λ G R. Hence, for any vector X 1 - tangent to N we have X 1 - f = 0. This together with (??) means that f is constant and therefore θ is parallel. Since the Weyl structure on M is not exact, we have f / 0 (otherwise θ = 0). Thus, any harmonic form on (M,g) is constant multiple of θ and so if b / 0, then b\ = 1. It follows from (??) that the Ricci curvature of N is positive. When n = 4, N is a compact simply connected 3-dimensional manifold admitting a metric of positive Ricci curvature and by a theorem of Hamilton (cf. [?] or Theorem 5.30 in [?]) N is diffeomorphic to S 3. When n = 3, N is a compact simply connected 2-dimensional manifold admitting a metric of positive Ricci curvature (i.e. of positive Gauss curvature) and therefore is diffeomorphic to S 2. Thus b) is proved. If the inequality (??) is strict for X = θ# at some point, then it follows from (??) that θ is not a harmonic form. This means that there are no harmonic 1-forms, which proves a). Q.E.D. We note that if M is 3-dimensional, the left-hand side of the condition (??) in Theorem?? may be non-positive. Corollary 2.2 Any oriented 3-dimensional manifold M, which satisfies the assumptions of Theorem?? and has b / 0, is diffeomorphic to S 1 x S 2, but not isometric to S 1 x S 2 with a product metric. Proof: By Theorem?? the universal cover of M is isometric to Rxff, where N is diffeomorphic to S 2. Any metric on S 2 is conformal to the standard one and therefore the orientation-preserving isometries of N are contained in SL(2,C). Now, by arguments similar to those in the proof of Theorem 3.2 in [?] we obtain that the fundamental group of M is isomorphic to Z and M is the total space of a locally trivial fibre bundle over S 1 with fibre S 2 and structure group contained in SL(2,C). Hence, M is diffeomorphic to S 1 x S 2. The manifold M could not be isometric to S 1 x S 2 with a product metric because in this case the 1-form θ would be exact, which contradicts the assumption that the Weyl structure is non-exact. Q.E.D. Proof of Theorem??. It follows at once from Theorem?? and Corollary??. Q.E.D. The following result is proved in [?] for Einstein-Weyl manifolds.

7 Corollary 2.3 If (M,c) is a compact 4-dimensional conformal manifold with a Weyl structure with the symmetric part of the Ricci tensor non-negative everywhere and positive at some point, then the 1-form θ given by any choice of a compatible metric must vanish somewhere. Proof: It follows from Theorem?? that b1 = 0 and therefore the Euler characteristic of M is positive, which proves the Corollary. Q.E.D. 3 Ricci-positive Hermit ian-weyl structures Let (M,g,J) be a 2m-dimensional (m > 1) almost Hermitian manifold with almost complex structure J and compatible metric g. Let Q be the Kahler form of (M,g, J), defined by n(x,y)=g(x,jy). Denote by θ the Lee form of (M,g, J), 9 = Jd*fl. m 1 (For a 1-form α we shall denote by Jα the form dual to Jα#, where α# is the vector dual to α. Equivalently, Jα = a o J.) The canonical Weyl structure V w corresponding to the almost Hermitian structure (g, J) on M is determined by the metric g and its Lee form θ. The canonical Weyl structure does not depend on the choice of the metric in the conformal class c of g, i.e. the Weyl structure determined by a metric jec and its Lee form 9 coincides with V w. The canonical Weyl structure is called Hermit ian-weyl [?] if it preserves J. Since V w is torsion-free, this shows that J must be integrable. According to the results in [?], the canonical Weyl structure on a Hermitian manifold (M,g, J) is Hermit ian-weyl iff (3.13) dq = 9AQ. This condition is always satisfied in the 4-dimensional case. For higher dimensions it means that the Hermitian manifold (M,g,J) is locally conformally Kahler. In particular, dθ = 0. Notice that since the antisymmetric part of the Ricci tensor of V w is ^d9, on locally conformally Kahler manifolds the Ricci tensor of the canonical Weyl connection is symmetric. A particular subclass of locally conformally Kahler manifolds is formed by the generalized Hopf manifolds - the Hermitian manifolds with non-zero Lee form which is parallel with respect to the Levi-Civita connection. Proof of Corollary??. It follows from Theorem?? that b1 < 1. By (??) we see that the non-exactness of the Weyl structure implies that the Gauduchon metric has non-negative nonidentically zero scalar curvature. Thus, by Gauduchon's plurigenera theorem all the plurigenera of (M,J) vanish, cf. Proposition I.18 in [?] or [?].

8 If b1 = 0 we have that the Hodge number h 0,1 = 0 (see e.g. [?]). Hence, by the Castelnuovo criterion (cf. [?]) (M, J) must be a rational surface. The assertion about c21 follows by the same arguments as those in the proof of Theorem 1.3 in [?]. If b1 = 1 then Theorem?? shows that (M, J) is a generalized Hopf manifold with respect to the Gauduchon metric. The vanishing of the plurigenera means that the Kodaira dimension of (M, J) is oo [?]. The classification of generalized Hopf surfaces [?] implies that the surface is a Hopf surface of class 1. Q.E.D. For the rest of this section we assume that (M,g, J) is a Hermitian manifold, such that the canonical Weyl connection V w is Hermitian-Weyl. The Chern connection V c of (M,g, J) is given by (3.14) #(v r, Z) = g{v x Y, Z) + Un(jx, Y, Z). Combining (??), (??) and (??) we get (3.15) V c = V w + -0<S>Id + -J0(g> J, Z Z where Id denotes the identity of TM. It follows from (??) that the curvature tensors R C R W of V c and V w are related by and (3.16) RC = R W + 1d(Jθ) eg) J + 1dθ eg) Id. z z Notice that for m > 2 the last term in (??) is absent, since in this case dθ = 0. Let R (n)(x,y) = -Y,9(R C (e i,je i )X,Y), 3 = 1 where {e1,..., e2 m } is an orthonormal frame of tangent vectors. We denote by k C J-invariant bilinear form associated to R C \Q) and defined by the symmetric, kc(x,y) = R (Q)(JX,Y). Using (??) and the fact that R W o J = J o R W we obtain (3.17) k C = Ric W + 1 <d(j6),n > g. This equality is proved in [?] (formula (14)) in dimension 4. The proof in higher dimensions is the same, using that dθ = 0. Proof of Theorem??: We shall use formula (2.8) in [?], according to which on every generalized Hopf manifold (3.18) d(jθ) = \e\ 2 n + 6A Jθ. The equalities (??) and (??) yield (3.19) kc = Ric W + (m-l)\0\ 2 g. 8

9 Since Ric W > 0 and θ = 0, equality (??) implies that k C > 0. This allows us to apply the vanishing theorem for holomorphic (p, 0)-forms [?,?] and we deduce that h p, = 0, p = 1,2,...,m. On the other hand, the Hodge numbers on a 2m-dimensional generalized Hopf manifold satisfy the following relations [?]: (3.20) h m, = h m - 1 ' 0, h,p = h p, + lf- l >, p<m-l, (3.21) /,i.o = l ( 6 l _ 1 ) ),h0,1 1. = (b1 + 1) From h 1,0 = 0 and (??) we get b 1 = 1, h 0,1 = 1. Further, (??) implies that h,p = 0, p = 2,...,m 1. Finally, the Kodaira-Serre duality and h m, = 0 lead to the vanishing of h,m, which completes the proof. Q.E.D. Notice that if 2m = 4 the result of Theorem?? follows immediately from Corollary??. Acknowledgments The authors are supported by Contract MM 809/1998 with the Ministry of Science and Education of Bulgaria and by Contract 238/1998 with the University of Sofia "St. Kl. Ohridski". The second author thanks to The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, where the final part of this work was done. Both authors thank to V. Apostolov for his valuable comments and suggestions. References [1] B. Alexandrov, G. Grantcharov, S. Ivanov An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting parallel one-form, J. Geom. Phys., 28 (1998), [2] B. Alexandrov, S. Ivanov, Vanishing theorems on Hermitian manifolds, mathdg/ [3] W. Barth, C. Peters, A. Van de Ven, Compact Complex Surfaces, Springer-Verlag, [4] F. Belgun, On the metric structure of non-kahler complex surfaces, Math. Ann., to appear. [5] A. Besse, Einstein manifolds, Springer-Verlag, New York [6] D. Calderbank, H. Pedersen, Einstein-Weyl geometry, Edinburgh Preprint MS (1998), to appear in Essays on Einstein manifolds, International Press. [7] S. Dragomir, L. Ornea, Local conformal Kdhler geometry, Progress in Mathematics 155, Birkhauser, [8] P. Gauduchon, Le theoreme de Vexcentricite nulle, C. R. Acad. Sci. Paris Ser. A 285 (1977),

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