Kähler manifolds with homothetic foliation by curves

Differential Geometry and its Applications 46 (2016) 119–131

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Differential Geometry and its Applications www.elsevier.com/locate/difgeo

Kähler manifolds with homothetic foliation by curves Włodzimierz Jelonek Institute of Mathematics, Cracow University of Technology, 31-155 Kraków, Warszawska 24, Poland

a r t i c l e

i n f o

Article history: Received 7 October 2013 Received in revised form 8 February 2016 Available online xxxx Communicated by J. Slovak

a b s t r a c t The aim of this paper is to classify compact, simply connected Kähler manifolds which admit totally geodesic, holomorphic complex homothetic foliations by curves. © 2016 Elsevier B.V. All rights reserved.

MSC: 53C55 53C25 Keywords: Kähler manifold Holomorphic foliation Homothetic foliation Special Kähler–Ricci potential Special Kähler potential

1. Introduction The aim of the present paper is to classify compact, simply connected Kähler manifolds (M, g, J), dim M = 2n > 2, admitting a global, complex homothetic foliation F by curves which is totally geodesic and holomorphic. A foliation F on a Riemannian manifold (M, g) is called conformal if LV g = α(V )g holds on T F ⊥ where α is a one form vanishing on T F ⊥ . A foliation F is called homothetic if is conformal and dα = 0 (see [16,3,15]). Homothetic foliations are closely related with harmonic morphisms [2,17,15]. Homothetic foliations are also helpful for integrating Einstein [17,5,6] and Einstein–Weyl equations [13,10]. Any foliation induced by a horizontally homothetic submersion is homothetic and any homothetic foliation is locally given as fibers of a horizontally homothetic submersion.

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.difgeo.2016.02.004 0926-2245/© 2016 Elsevier B.V. All rights reserved.

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Complex homothetic foliations by curves on Kähler manifolds were recently classified locally in [3]. In [3] the authors describe a homothetic totally geodesic foliation F on the total space without zero section L× of a holomorphic line bundle p : L → M over Kähler manifold N whose curvature is iωN , where ωN is the Kähler form of N . The metric on L× is given by the Calabi construction [8]. L× admits a Hamiltonian Killing field X and the foliation is tangent to the distribution spanned by {X, JX}. Next the authors using holomorphic twists give local description of complex homothetic foliations by curves. The manifold L× is clearly noncompact and the obtained results cannot be used to the global classification. In our paper we investigate the global structure of compact, simply connected Kähler manifolds with complex, holomorphic, totally geodesic foliation by curves. The result is complementary to [3], were the local structure of Kähler manifolds with complex homothetic foliation was studied. We classify compact simply connected Kähler manifolds admitting a global, complex homothetic foliation F satisfying the conditions U = ∅, where U = {x ∈ M : |α| = 0}, i.e. if α does not vanish identically. In particular we show that such foliation is given as the fibers of horizontally homothetic submersion onto a Hodge manifold. First we show that (M, g, J) admits a global holomorphic Killing vector field with a Killing potential, which is a special Kähler potential τ . This Killing field is an eigenfield of the Ricci tensor of (M, g, J). The dilation of the foliation F is |τ1−c| and α = d ln |τ − c| for a certain constant c. Next we use slightly generalized results of Derdzinski and Maschler [5,6]. Our results rely heavily on the papers [5,6]. As a corollary we prove that every compact simply connected Kähler manifold admitting a holomorphic totally geodesic homothetic foliation, with αx0 = 0 at least at one point x0 ∈ M , is a holomorphic CP1 -bundle onto a Hodge manifold (N, h, J), M = P(L ⊕ O), where L is a holomorphic line bundle with curvature form Ω = sΩN , where s = 0 and ΩN is the Kähler form of (N, h, J). The leaves of the foliation are the fibers CP1 of the bundle. The distribution associated to the foliation F|U is V = span{∇τ, J∇τ }. If α = 0 then M is a product of a Riemannian surface and a Kähler manifold. The result was partially proved in [9] and for the completeness we cite some of results from [9] in our present paper.

2. Principal field

Let (M, g, J) be a 2n-dimensional Kähler manifold with a 2-dimensional J-invariant distribution D determined by a complex foliation F. Let X(M ) denote the algebra of all differentiable vector fields on M and Γ(D) denote the set of local sections of the distribution D. If X ∈ X(M ), then by X  we shall denote the 1-form φ ∈ X∗ (M ) dual to X with respect to g, i.e. φ(Y ) = X  (Y ) = g(X, Y ). By Ω we shall denote the Kähler form of (M, g, J), i.e. Ω(X, Y ) = g(JX, Y ). Let us denote by E the distribution D⊥ , which is a 2(n − 1)-dimensional J-invariant distribution. By h, m respectively we shall denote the tensors h = g ◦ (pD × pD ), m = g ◦ (pE × pE ), where pD , pE are the orthogonal projections on D, E respectively. It follows that g = h + m. By ω we shall denote the Kähler form of D i.e. ω(X, Y ) = h(JX, Y ) and by Ωm the Kähler form of E i.e. Ωm (X, Y ) = m(JX, Y ). For any local section X ∈ Γ(D) we define divE X = trm ∇X  = mij ∇ei X  (ej ) where {e1 , e2 , . . . , e2(n−1) } is any basis of E and [mij ] is a matrix inverse to [mij ], where mij = m(ei , ej ). Note that if f ∈ C ∞ (M ) then divE (f X) = f divE X in the case X ∈ Γ(D). Let ξ ∈ Γ(D) be a unit local section of D. Then {ξ, Jξ} is an orthonormal basis of D. Let η(X) = g(ξ, X) and Jη = −η ◦ J which means that Jη(X) = g(Jξ, X). Let us denote by κ the function  κ = (divE ξ)2 + (divE Jξ)2 . The function κ does not depend on the choice of a section ξ. It turns out that κ = (n − 1)|α| if F is a complex, homothetic foliation. Hence κ is a well defined, continuous function on M , which is smooth in the open set U = {x : κ(x) = 0}. There is a smooth, global unit section ξ ∈ Γ(U, D) defined on U uniquely up to a sign such that divE Jξ = 0. The section ξ we shall call the principal section of D (see also [7]). Note that divE ξ = κ.

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3. Complex homothetic foliations We start with (see [16,2,3,15]): Definition. A foliation F on a Riemannian manifold (M, g) is called conformal if LV g = α(V )g holds on T F ⊥ where α is a one form vanishing on T F ⊥ . A foliation F is called homothetic if it is conformal and dα = 0. In the rest of the paper we assume that dim F = 2 and F is complex, which means that for an associated distribution D we have JD = D. Let us write α(X) = g(ζ, X). Then divE Jζ = 0, divE ζ = (n − 1)|α|2 , 1 which means that in U the field ξ = |α| Jζ is the principal field. Let η = ξ  . Note that κ = (n − 1)|α|. If we assume that F is totally geodesic, then dE |α| = 0 i.e. d|α|(X) = 0 if X ∈ E. In fact [3] since dα = 0 we get Xg(ζ, ζ) = 2g(∇X ζ, ζ) = 2g(∇ζ ζ, X) = 0. A distribution D is called holomorphic if LX JT M ⊂ D for any X ∈ Γ(D). Hence if D is holomorphic, then for any X ∈ Γ(D), Y ∈ X(M ), Z ∈ Γ(E) we have g(LX JY, Z) = 0. Let A = ∇X. Note that LX J(Y ) = A ◦ JY − J ◦ AY = [A, J]Y . Consequently g(AJY, Z) = g(JAY, Z)

(3.1)

for Y ∈ X(M ), Z ∈ Γ(E). Let us write ∇Jξ Jξ = p∗ ξ for a certain function p∗ ∈ C ∞ (U ). Now we shall show that totally geodesic, holomorphic complex homothetic foliations by curves are closely related with the so called B0 -distributions introduced in [7]. It enables us later to use the results from [9] to describe them. Proposition 3.1. Let a foliation F on a Kähler manifold (M, g, J) be totally geodesic, holomorphic complex homothetic foliation by curves. Then in U dη = 0, ∇ξ ξ = 0, d ln |α| = −(|α| + p∗ )η, dp∗ ∧ η = 0, ∇X η(Y ) =

1 |α|m(X, Y ) − p∗ Jη(X)Jη(Y ). 2

Proof. The distribution Δ = {X ∈ T U : η(X) = 0} defined in U is integrable. From (3.1) it follows that ∇η(JX, JY ) = ∇η(X, Y ) for X, Y ∈ Γ(E). If we take X = ξ, A = ∇ξ in (3.1), then we obtain g(AX, Y ) + g(X, AY ) = α(ξ)g(X, Y ) for any X, Y ∈ Γ(E). On the other hand, since dα = 0, we get g(AX, Y ) = g(X, AY ) on E. Hence g(AX, Y ) = 1 2 |α|g(X, Y ). Consequently, since D is totally geodesic, we get ∇X η(Y ) =

|α| m(X, Y ) + pη(X)Jη(Y ) − p∗ Jη(X)Jη(Y ), 2

(3.2)

where ∇ξ ξ = pJξ. It is also clear that dη = pη ∧ Jη and dJη = |α|Ω − (p∗ + |α|)η ∧ Jη. Thus the distribution E|U is the so called B0 -distribution defined in [7]. If dim M = 2n ≥ 4 we also have p = g(∇ξ ξ, Jξ) = 0 and consequently ∇ξ ξ = 0, dη = 0. In fact we get dp ∧ η ∧ Jη + |α|pη ∧ Ω = 0 and consequently p = 0 in U = {x ∈ M : |α| = 0}. From the above equations we obtain (d|α| + |α|(|α| + p∗ )η) ∧ Ω = d(p∗ + |α|) ∧ η ∧ Jη.

(3.3)

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1 1 Since d|α| ∈ D we get (d|α| + |α|(|α| + p∗ )η) ∈ D. Thus it follows from (3.3) that d|α| + |α|(|α| + p∗ )η = 0. Hence d ln |α| = −(|α| + p∗ )η, dp∗ ∧ η = 0. Thus ∇X η(Y ) = 12 |α|m(X, Y ) − p∗ Jη(X)Jη(Y ).

(3.4)

2

4. Examples and Killing vector fields with special Kähler potential First we give a definition Definition. A nonconstant function τ ∈ C ∞ (M ), where (M, g, J) is a Kähler manifold, is called a special Kähler potential if the field X = J(∇τ ) is a Killing vector field and, at every point with dτ = 0, all non-zero tangent vectors orthogonal to the fields X, JX are eigenvectors of ∇dτ . Let τ be a special Kähler potential on a Kähler manifold (M, g, J), V = span{∇τ, J∇τ } on U = {x ∈ M : dτ (x) = 0} and let F be a foliation on U given by the integrable distribution V. Then F Θ Θ is a totally geodesic, holomorphic complex conformal foliation. We have α = 2 Q dτ , ζ = 2 Q ∇τ , where by Θ τ ⊥ we denote the eigenvalue of the Hessian H corresponding to the distribution H = V and Q = g(∇τ, ∇τ ). By Λ we denote the eigenvalue of the Hessian H τ corresponding to the distribution V. Proposition 4.1. Let X = J(∇τ ) be a holomorphic Killing field on a Kähler manifold (M, g, J). Then X is an eigenfield of the tensor H τ if and only if dQ = 2Λdτ

(4.1)

for a certain function Λ, which is then an eigenfunction of H τ . Proof. Since X is a holomorphic Killing field then ddc τ (Y, Z) = 2H τ (JY, Z), since ddc = 2i∂∂. Note that LX dc τ = −LX (dτ ◦ J) = LX (dτ ) ◦ J = d(LX τ ) ◦ J = 0. Hence Xddc τ = −d(Xdc τ ) = −d(g(∇τ, ∇τ )) = −dQ. On the other hand ddc τ (Y, Z) = 2H τ (JY, Z), hence −dQ = 2H τ (JX, .) = −2H τ (∇τ, .). The field ∇τ is an eigenfield of H τ if and only if H τ (∇τ, .) = Λg(∇τ, .) = τ Λdτ . It follows that ∇τ is an eigenfield of H τ if dQ = 2Λdτ and then H|V = ΛidV . 2 Proposition 4.2. F is a totally geodesic, holomorphic complex homothetic foliation by curves if and only if the field X = J(∇τ ) is an eigenvector of the Ricci tensor ρ of (M, g, J). Θ Proof. We have to show that dα = 0 where α = 2 Q dτ . Since dQ = 2Λdτ it is equivalent to dΘ ∧ dτ = 0. On the other hand dΔτ = 2ρ(∇τ, .) since ∇τ is holomorphic and Δτ = −g(g, ∇dτ ) = −(2Λ + 2(n − 1)Θ). Hence dΘ ∧ dτ = 0 if and only if ρ(∇τ, .) ∧ dτ = 0, which means that ∇τ is an eigenfield of the Ricci tensor ρ. 2

Remark. It follows that compact Kähler manifolds admitting special Kähler–Ricci potential described in [5], which are holomorphic CP1 bundles over Kähler Einstein manifolds, give examples of totally geodesic, holomorphic complex homothetic foliation by curves. The leaves of the foliation are the fibers CP1 of the bundle.

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If a Killing field X has a special Kähler potential, then the distribution V = span{X, JX} is totally geodesic. In fact ∇X X = ΛJX = −∇JX JX and ∇X JX = J∇X X = −ΛX. Now we prove that if dim M ≥ 6 then a Killing field X with a special Kähler potential is in U an eigenvector of the Ricci tensor S of (M, g, J). This fact does not hold if dim M = 4 as Derdzinski shows in [4]. Let us recall that a dilation of conformal foliation F is a function φ such that α = −2d ln φ. Theorem 4.3. Let X = J(∇τ ) be a holomorphic Killing field with a special Kähler potential τ on a Kähler manifold (M, g, J) and dim M ≥ 6. Then SX = λX in U , where λ ∈ C ∞ (U ) and d(Δτ ) = 2λdτ , [S, T ] = 0 and ∇X S = 0 where T = ∇X, dQ = 2Λdτ , F is in U a homothetic, holomorphic, totally geodesic foliation. If Θ = 0 at least at point x0 ∈ U then in U F has the dilation |τ1−c| , Q Θ = 2(τ − c) and α = d ln |τ − c| for some constant c ∈ R. (f) If η(Y ) = g(X, Y ) = dc τ then Δη = 2ρ(X, .) = 2λη, where ρ is the Ricci tensor, and ddc τ = 2ΛωV + 2ΘωH . 2 1d Q (g) The sectional curvature of the leaves of the foliation F equals K(X ∧ JX) = − dΛ dτ = − 2 dτ 2 .

(a) (b) (c) (d) (e)

Proof. Note first that H τ (Y, Z) = g(∇Y ∇τ, Z) = −g(J∇Y J∇τ, Z) = −g(JT Y, Z)

(4.2)

1 and functions Λ, Θ are smooth in U . We have 2Λ + 2(n − 1)Θ = −Δτ and Λ = Q H τ (X, X) ∈ C ∞ (U ). Note that JT X = −ΛX, which implies J∇Y T X + JT 2 Y = −Y ΛX − ΛT Y . Consequently

g(JR(Y, X)X, JY ) − g(T Y, T Y ) = −Y Λg(X, JY ) − Λg(T Y, JY ), R(Y, X, X, Y ) − ||T Y ||2 = Y ΛY τ + Λg(JT Y, Y ) where we assume that R(X, Y, Z, W ) = g(R(X, Y )Z, W ) for any X, Y, Z, W ∈ T M . In particular if Y ∈ H, then R(Y, X, X, Y ) = Θ2 ||Y ||2 − ΛΘ||Y ||2 = Θ(Θ − Λ)||Y ||2 . On the other hand 2T X = −∇Q and R(Y, X)Y + T 2 Y = − 12 ∇Y ∇Q, which implies R(Y, X, X, Y ) = ||T Y ||2 − 12 H Q (Y, Y ). Hence R(Y, X, X, Y  ) = g(T Y, T Y  ) − 12 H Q (Y, Y  ). For Y ∈ H we get 12 H Q (Y, Y ) = ΛΘ||Y ||2 . Hence 1 1 λQ = 2Λ2 + (2n − 2)Θ2 + ΔQ, H Q (X, X) = Λ2 ||X||2 , 2 2 1 Q H (JX, JX) = Λ2 ||X||2 − R(JX, X, X, JX). 2 If Y ∈ H, then 12 H Q (Y, Y ) = ΛΘ||Y ||2 and R(Y, X, X, Y ) = Θ(Θ − Λ)||Y ||2 . 1 Note that − 12 ΔQ = −(2Λ2 + (2n − 2)ΘΛ) + Q R(JX, X, X, JX) = λQ − (2Λ2 + (2n − 2)Θ2 ) and 1 K(X ∧ JX) = Q12 R(JX, X, X, JX) = λ + 2(n−1) Θ(Λ − Θ), where λ = Q ρ(X, X). Q Now we show that the function Λ, Θ satisfy equations dΛ ∧ dτ = 0 = dΘ ∧ dτ if dim M ≥ 6 which is equivalent to the fact that X is an eigenfield of the Ricci tensor S. Let ω1 = ω V , ω2 = ω H , where

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ω is the Kähler form of (M, g, J). If η(Y ) = g(X, Y ), η(Y ) = −g(JX, Y ) then η = dc τ and we get 1 dη(Y, Z) = 2g(T Y, Z) and (f) follows. Note that ω1 = Q η ∧ η. On the other hand dη = 2Λω1 + 2Θω2 , dη = 0, dω1 = −dω2 , ∇ω1 = −∇ω2 .

(4.3)

1 1 1 Hence dω1 = − Q η ∧ dη = −2 Q (Λη ∧ ω1 + Θη ∧ ω2 ). We also have dΛ ∧ ω1 − 2(Λ − Θ) Q (Λη ∧ ω1 + Θη ∧ ω2 ) + 1 dΘ ∧ ω2 = 0. Consequently dΛ ∧ ω1 + (dΘ − 2 Q (Λ − Θ)Θη) ∧ ω2 = 0. Thus ∇Λ ∈ Γ(V) and this relation 1 1 remains true also for dim M = 4 and dΘ = −2 Q (Θ − Λ)Θη which means that dΘ = 2 Q (Λ − Θ)Θdτ . We have 2Λ + 2(n − 1)Θ = −Δτ and consequently 2dΛ + 2(n − 1)dΘ = −dΔτ = −2λdτ , where Δτ = − trg H τ and λ is an eigenvalue of the Ricci tensor corresponding to X. Hence

QdΘ = 2Θ(Λ − Θ)dτ,

(4.4)

QdΛ = (2(n − 1)Θ(Θ − Λ) − λQ)dτ.

(4.5)

If we write dΛ = −qdτ then it follows from (4.5) that q = K(X ∧ JX). There exists a constant c such that Q Q dQ dΘ Λ Θ = 2(τ − c) or Θ = 0 in U . In fact if V = {x ∈ U : Θ(x) = 0} = ∅ then in V d Θ = Θ − Q Θ2 = 2 Θ dτ + Λ  )dτ = 2dτ which implies d( Q 2(1 − Θ Θ − 2τ ) = 0 and consequently V = U . Since the set M = {x : X(x) = 0} is connected we obtain Q = 2(τ − c). Θ

(4.6)

α = d ln |τ − c|.

(4.7)

Hence if α is not identically 0, then

which proves (e). We can assume that c = 0 replacing τ by τ − c.

2

Remark. In the case dim M = 4 we have to assume, that X is an eigenfield of the Ricci tensor S to obtain the above relations. In fact if n = 2 and X is an eigenfield of the Ricci tensor we have 2dΛ + 2(n − 1)dΘ = −dΔτ = −2λdτ , where Δτ = − trg H τ . Note that T X = ΛJX. Hence div(T X) = tr{Z → ∇Z (T X)} = tr{R(Z, X)X + T 2 Z} = g(SX, X) − ||T ||2 = λQ − ||T ||2 . On the other hand div(ΛJX) = tr{Z → ZΛJX + ΛJT Z} = dΛ(JX) − Λ(2Λ + 2(n − 1)Θ). Consequently we get λQ − 2Λ2 − 2(n − 1)Θ2 = dΛ(JX) − 2Λ2 − 2(n − 1)ΛΘ. Hence dΛ(JX) = −2(n − 1)Θ(Θ − Λ) + λQ which means that λQ = qQ + 2(n − 1)Θ(Θ − Λ), and then dΛ(JX) = −dΛ(∇τ ) = qQ. Hence qQ = −2(n − 1)Θ(Θ − Λ) + λQ. Consequently QdΛ = −Qqdτ = (2(n −1)Θ(Θ −Λ) −λQ)dτ . We get (n −1)QdΘ = (−2(n −1)Θ(Θ −Λ) +λQ −λQ)dτ = −2(n −1)Θ(Θ −Λ)dτ . The next result we shall need in the last section of the paper. Let Y ∈ H, then dτ (Y ) = dτ (JY ) = 0. Hence for any Z ∈ X(M ) we get ∇Z dτ (Y ) + dτ (∇Z Y ) = 0, ∇Z dτ (JY ) + dτ (J∇Z Y ) = 0, which implies ∇Z Y V = −

1 1 (H τ (Z, Y )∇τ − H τ (Z, JY )J∇τ ) = −Θ (g(Z, Y )∇τ + ω(Z, Y )J∇τ ). Q Q

Θ If we assume that Z ∈ H then [Z, Y ]V = 2 Q ω(Y, Z)J∇τ . Now it is easy to see just as in [5] that

QR(Y, Z)∇τ = 2(Λ − Θ)Θω(Y, Z)X, where we assume that R(u, v)w = ∇u ∇v w − ∇v ∇u w − ∇[u,v] w (a different notation then in [5]).

(4.8)

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Proposition 4.4. Let X = J(∇τ ) be a holomorphic Killing field on a Kähler manifold (M, g, J). Then in U = {x : Xx = 0} the following conditions are equivalent: (a) (b) (c) (d) (e)

X is an eigenfield of H τ , v = ∇τ is pregeodesic i.e. ∇v v = Λv, The distribution V = span(X, JX) is totally geodesic, The distribution V = span(X, JX) is an eigendistribution of H τ , dQ = 2Λdτ .

Proof. (a) ⇔ (b) X is an eigenfield of H τ if and only if dQ = 2Λdτ . We have ∇Q = −2∇X X = 2∇v v which means that ∇Q = 2Λ∇τ if and only if ∇v v = Λv. (b) ⇒ (c) Since ∇X X = ∇v v = Λv, then ∇X v = −J∇X X = −ΛJv, [X, JX] = 0 and we are done. (c) ⇒ (a) Since Xg(X, X) = 0 then g(∇X X, X) = 0 and ∇X X = λJX which means that ∇v v = −λv. The equivalence (a) ⇔ (d) is obvious. 2 If dQ = 2Λdτ then ΔQ = −2qQ + 2ΛΔτ . In fact ∇X ∇Q = 2XΛ∇τ + 2Λ∇X ∇τ and H Q (X, Y ) = −2qdτ (X)dτ (Y ) + 2ΛH τ (X, Y ). Q τ In particular H|V = Θg if and only if H|V = 2ΘΛg. Note that η = dc τ and

ddc τ = 2 Consequently if we denote θ =

s 2Q g(X, .)

=

Λ dτ ∧ dc τ + 2Θω H . Q

s c 2Q d τ

where s ∈ R, then dθ =

s H 2|τ −c| ω .

5. Local holomorphic Killing vector field on U Let (M, g, J) be a Kähler manifold of dimension 2n ≥ 4 admitting a global, complex homothetic foliation F by curves, which is totally geodesic and holomorphic. We shall show in this section using the ideas from [9] and Proposition 3.1 that for every x ∈ U there exists an open neighborhood V ⊂ U of x and a function f ∈ C ∞ (V ) such that XV = f Jξ is a Killing vector field in V , which we shall call a local special Killing vector field (see also [3]). Let V be a geodesically convex neighborhood of x in U . Then V is contractible. Note that the form φ = −p∗ η, where p∗ = g(∇Jξ Jξ, ξ) is closed in U , since by (3.4), dφ = −dp∗ ∧ η = 0. Consequently there exists a function F ∈ C ∞ (V ) such that dF = φ = −p∗ η.

(5.1)

Let f = exp ◦F and φ = f ξ  . Now it is clear (see [9, p. 147]) that ∇X φ(Y ) = ∇X (f η)(Y ) = f

|α| m(X, Y ) − f p∗ h(X, Y ) 2

(5.2)

and ∇X (f η)(Y ) = ∇Y (f η)(X). Consequently d(f η) = 0. It follows that there exists a function τ ∈ C ∞ (V ) such that f ξ = ∇τ . Consequently XV = f Jξ = J(∇τ ), which means that XV is a holomorphic Killing vector field with a Kähler potential τ . Let c : [0, l] → M be a unit geodesic such that c([0, l)) ⊂ U and c(l) ∈ K = {x ∈ M : κ(x) = 0}. A vector field C, which is a Jacobi field along c i.e. ∇2c˙ C − R(c, ˙ C)c˙ = 0, will be called a special Jacobi field if there exists an open, geodesically convex neighborhood V of c(0) such that C(0) = XV (c(0)),

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∇c˙ C(0) = ∇c˙ XV (c(0)). If im c ∩ V = c([0, )) then it follows that XV (c(t)) = C(t) for all t ∈ [0, ). We have the following lemma: Lemma 5.1. Let us assume that a vector field C along a geodesic c is a special Jacobi field along c. Then limt→l |C(t)| = 0, and g(c, ˙ C) = 0. ˙ = 0 since this property is valid for Killing vector fields. It follows that the Proof. Let us note that g(∇c˙ C, c) function g(c, ˙ C) is constant. Let k ∈ (0, l). Then c([0, k]) ⊂ U . For every t ∈ [0, k) there exists a geodesically convex open neighborhood Vt of the point c(t) and a special Killing vector field XVt = ft Jξ on Vt defined in Section 4. The field XVt is defined uniquely up to a constant factor. From the cover {Vt} : t ∈ [0, k] of the compact set c([0, k]) we can choose a finite subcover {Vt1 , Vt2 , . . . , Vtm }. Let ci be the part of geodesic c contained in Vi = Vti , i.e. im c ∩ Vi = im ci = c((ti , ti+1 )). We define the Killing vector field Xi in every Vi = Vti by induction in such a way that X1 = XV on V1 ∩ V and Xi = Xi+1 on Vi ∩ Vi+1 . Let Xi = fi Jξ. Note that C(t) = Xi ◦ c(t) = fi Jξ ◦ c(t) for t ∈ (ti , ti+1 ). Consequently, on Vi , |C| = fi . From (3.4) and (5.1) it follows that d ln κ = d ln fi −

κ η. n−1

Hence d κ d ln κ ◦ c(t) = ln |C(t)| − η(c(t)), ˙ dt dt n−1

(5.3)

κ ◦ c(t) κ d ln =− η(c(t)). ˙ dt |C(t)| n−1

(5.4)

and

Consequently κ ◦ c(0) 1 κ ◦ c(k) − ln =− ln |C(k)| |C(0)| n−1

k κη(c(t))dt. ˙

(5.5)

0

Hence 1 κ ◦ c(0) + ln |C(k)| = ln κ ◦ c(k) − ln |C(0)| n−1

k κη(c(t))dt. ˙

(5.6)

0

Note that k |

k κη(c(t))dt| ˙ ≤

0

k |κη(c(t))|dt ˙ ≤

0

k κ|c(t)|dt ˙ ≤

0

κdt.

(5.7)

0

Let κ0 = sup{κ(x) : x ∈ c([0, l])}. From (5.6) it follows that ln |C(k)| ≤ ln κ ◦ c(k) − ln Consequently

1 κ ◦ c(0) + κ0 l. |C(0)| n−1

(5.8)

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lim sup ln |C(k)| ≤ lim ln κ ◦ c(k) − ln k→l−

k→l−

1 κ ◦ c(0) + κ0 l = −∞. |C(0)| n−1

127

(5.9)

From (5.9) it is clear that limk→l− |C(k)| = 0. Since |g(c(t), ˙ C(t)) ≤ |C(t)| and g(c, ˙ C) is constant it follows that |g(c(t), ˙ C(t))| ≤ limt→l |C(t)| = 0 which means that g(c, ˙ C) = 0. 2 6. Global holomorphic Killing vector field on M From now on we assume that (M, g, J) is a complete Kähler manifold with totally geodesic, holomorphic, complex homothetic foliation F, dim M ≥ 4 and the set U = {x ∈ M : |α|(x) = 0} is non-empty. Let K = {x ∈ M : |α|(x) = 0}. Theorem 6.1. The set U is connected and the set K has an empty interior. Proof. We use Lemma 5.1. See [9, p. 149].

2

Lemma 6.2. On every geodesically convex open set V in M a holomorphic Killing vector field X can be defined such that for every open geodesically convex set W ⊂ V ∩ U the restriction X|W is a special Killing vector field on W . Proof. See [9].

2

Let V1 , V2 be two open, geodesically convex sets. The fields X1 , X2 which exist on V1 , V2 respectively by Lemma 6.2 satisfy Xi|K∩Vi = 0. What is more for some constant C12 equation X1 = C12 X2 holds on V1 ∩ V2 . Now we have Theorem 6.3. Let (M, g, J) be a complete Kähler manifold of dimension 2n ≥ 4 with totally geodesic, holomorphic complex homothetic foliations by curves. Let αx0 = 0 at least at one point x0 ∈ M . If H 1 (M, R) = 0 then there exists on M a non-zero holomorphic Killing vector field X = J(∇τ ) with a special Kähler potential τ such that X is an eigenfield of the Ricci tensor of (M, g, J). Proof. See [9]. Thus X = J(∇τ ) is a holomorphic Killing vector field with a Killing potential τ , which restricted to a geodesically convex set V ⊂ U , is a special Killing vector field. Note that in view of (5.2) the Killing field X constructed by us is a Killing vector field with a special Kähler potential τ , which is an eigenfield of the Ricci tensor by Proposition 4.2 and Theorem 4.3. 2 Corollary 6.4. Let (M, g, J) be a complete Kähler manifold of dimension 2n ≥ 4 with totally geodesic, holomorphic complex homothetic foliations by curves. Let αx0 = 0 at least at one point x0 ∈ M and let ˜ , g˜) be the Riemannian universal covering space of (M, g, J). Then there exists on (M ˜ , g˜) a non-zero (M holomorphic Killing vector field X with a special Kähler potential such that X is an eigenfield of the Ricci ˜ , g˜). tensor of (M Remark. Note that if dim M ≥ 6 then for every special Kähler potential τ the vector field J(∇τ ) is automatically eigenfield of the Ricci tensor S of (M, g) (see Theorem 4.3). If dim M = 4 then from Theorem 6.3 it follows that in our case τ is a special Kähler–Ricci potential [5].

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128

7. Construction of Kähler manifolds Let (N, h, J) be a simply connected Hodge manifold, i.e. (N, h, J) is a Kähler manifold and the cohomols ogy class { 2π ΩN } is an integral class, where ΩN is the Kähler form of (N, h, J) and s ∈ R. We also assume that dim N = 2m > 2. Let s > 0, L > 0, L ∈ R and r : [0, L] → R be a positive smooth function on [0, L] with r (t) > 0 for t ∈ (0, L), which is even at 0 and L, i.e. there exists an  > 0 and even, smooth functions r1 , r2 : (−, ) → R such that r(t) = r1 (t) for t ∈ [0, ) and r(t) = r2 (L − t) for t ∈ (L − , L]. Then it is clear that the function f = 2s rr is positive on (0, L) and f (0) = f (L) = 0. Let P be a circle bundle over s N classified by the integral cohomology class { 2π ΩN }. On the bundle p : P → N there exists a connection ∗ form θ such that dθ = sp ΩN where p : P → N is the bundle projection. Let us consider the manifold (0, L) × P with the metric (see [1,14]) g = dt2 + f (t)2 θ2 + r(t)2 p∗ h,

(7.1)



if s = 0. The metric 6.1 is Kähler if and only if f = 2rr s (see [9, p. 155]). This time we do not assume that (N, h, J) is Einstein. It is known that the metric (7.1) extends to a metric on a sphere bundle M = P ×S 1 CP1 if and only if the function r is positive and smooth on (0, L), even at the points 0, L, the function f is positive, smooth and odd at the points 0, L and additionally f  (0) = 1, f  (L) = −1. If f =

2rr  s

(7.2)

for r as above, then (7.2) means that 2r(0)r (0) = s, 2r(L)r (L) = −s.

(7.3)

Let L be a complex line bundle L = P ×S 1 C with the Hermitian fiber metric <, > induced by standard Hermitian metric on C. Then P is a principle bundle of unit frames on L and θ is a connection on P making <, > parallel. Note that L − {0N } = P × R+ where R+ = {r ∈ R : r > 0}. The horizontal distribution of the connection on L induced by θ and making <, > parallel coincides with H = ker θ ⊂ T P ⊂ T P ⊕ T R+ . In [5, §4], there is constructed a metric on L given by g = 2|τ − c|p∗ h on H, g = (ar)−2 Q <, > on V, g(H, V) = 0, Q where V ⊂ T L is a vertical bundle. The function Q of r ∈ R+ satisfies the equation dτ dr = ar where Q 2 2 2 a ∈ R − {0}. Note that the metric g on V is g = a2 r2 (dr + r θ ). If we change the coordinate r by t on R+ ar √ such that dr then the metric on V has the form g = dt2 + aQ2 θ2 . Then the metric on L is dt = Q

g = dt2 +

Q 2 θ + 2|τ − c|p∗ h a2

where θ is the connection form. 8. Circle bundles s Let (N, h, J) be a Hodge manifold with integral class { 2π ΩN }, where s ∈ R, s > 0 and let p : P → N be a circle bundle with a connection form θ such that dθ = sΩN (see [11], [9, pp. 152–154]). Let us assume that dim N = 2m = 2(n − 1). Let us consider a Riemannian metric g on P given by

g = a2 θ ⊗ θ + b2 p∗ h

(8.1)

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129

where a, b ∈ R. Let ξ be the fundamental vector field of the action of S 1 on P i.e. θ(ξ) = 1, Lξ g = 0. It follows that ξ ∈ iso(P ) and a2 θ = g(ξ, .). Consequently a2 dθ(X, Y ) = 2g(T X, Y ) for every X, Y ∈ X(P ), where T X = ∇X ξ. Note that g(ξ, ξ) = a2 is constant, hence T ξ = 0. On the other hand dθ(X, Y ) = ˜ = 0 and sp∗ ΩN (X, Y ) = sh(Jp(X), p(Y )). Note that there exists a tensor field J˜ on P such that Jξ H H H ˜ J(X) = (JX∗ ) where X = X∗ ∈ T P is the horizontal lift of X∗ ∈ T N (i.e. θ(X∗ ) = 0) and X∗ = p(X). Since T ξ = 0 we get ∇T (X, ξ) + T 2 X = 0 and R(X, ξ)ξ = −T 2 X. Thus g(R(X, ξ)ξ, X) = ||T X||2 and 4 ρ(ξ, ξ) = ||T ||2 = sa 4b4 2m. Consequently 1 ξ ξ s2 a2 λ = ρ( , ) = 2 ||T ||2 = m. a a a 2b4 We shall compute the O’Neill tensor A (see [12, p. 460]) of the Riemannian submersion p : (P, g) → (N, b2 h). We have AE F = a12 (g(E, T F )ξ + g(ξ, F )T E). If U, V ∈ H then ||AU V ||2 =

1 s2 a2 2 ˜ )2 . g(E, T F ) = g(E, JF a2 4b4

If E is horizontal and F is vertical then AE F = a12 g(ξ, F )T E. Hence AE ξ = T E and ||AE ξ||2 = ||T E||2 = s2 a4 s2 a2 4b4 . It follows that K(PEξ ) = 4b4 , where K(PEF ) denotes the sectional curvature of the plane generated by vectors E, F . If E, F ∈ H then (see [9, p. 153], [12, p. 465]) K(PEF ) =

˜ )2 1 3s2 a2 g(E, JF K0 (PE∗ F∗ ) − , 2 4 b 4b ||E ∧ F ||2

where K0 stands for the sectional curvature of the metric h on N . Applying this we get for any E ∈ H the 2 2 formula for the Ricci tensor ρ of (M, g): ρ(E, E) = b12 ρ0 (bE∗ , bE∗ ) − s2ba4 , where ρ0 is a Ricci tensor of (M, h). Now we shall find a formula for R(X, ξ)Y where X, Y ∈ H. We have R(X, Y, Z, ξ) = 0 for X, Y, Z ∈ H, and 2 4 R(X, ξ, Y, ξ) = − s4ba2 h(X∗ , Y∗ ). Hence ρ(ξ, X) = 0 if g(X, ξ) = 0 and ξ is an eigenfield of ρ. Note that the Ricci tensor ρ has at a point x ∈ P k + 1 eigenvalues where k is the number of eigenvalues of ρ0 at a point p(x). 9. A global classification of compact Kähler manifolds with homothetic foliation by curves Now we study Riemannian submersion p : (0, L) × P → (0, L). In this case the O’Neill tensor A = 0. We shall compute the O’Neill tensor T (see [12, p. 460]). Note that, since Lξ θ = 0, the field ξ is the Killing vector field for ((0, L) × P, g), where g = dt2 + f (t)2 θ2 + r(t)2 p∗ h. We show that (0, L) × P is a Kähler manifold 2 and ξ is a Killing vector field with the special Kähler potential rs . Let us denote by Y ∗ the horizontal lift of the vector Y ∈ T N with respect to the Riemannian submersion pN : P → N i.e. pN (Y ∗ ) = Y, g(Y ∗ , ξ) = 0. d Let H = dt be the horizontal vector field for this submersion and D be the distribution spanned by the vector fields H, ξ. If U, V ∈ V (here V temporary denotes the vertical distribution of the above Riemannian submersion) and g(U, V ) = 0 then T (U, V ) = 0. Let U ∈ V, g(U, ξ) = 0 and U = U∗∗ with h(U∗ , U∗ ) = 1, then the following formula holds T (U, U ) = −rr H. In fact 2g(∇U V, H) = −Hg(U, V ) = −2rr h(U∗ , V∗ ) if U = V or 0 if g(U, V ) = 0. We also have T (ξ, ξ) = −f f  H. The almost complex structure defined by JH = f1 ξ, JX = (J∗ X∗ )∗ for X = (X∗ )∗ ∈ E = D⊥ where X∗ ∈ T N , is a Kähler structure with respect to the metric g. The proof is similar to that in [9] although now we do not assume that (N, h) is Einstein. Let U, V, W ∈ V and g(U, ξ) = g(V, ξ) = g(W, ξ) = 0. Then for R(X, Y, Z, W ) = g(R(X, Y )Z, W ) we get

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130

ˆ R(U, V, ξ, W ) = R(U, V, ξ, W ) + g(T (U, ξ), T (V, W )) − g(T (V, ξ), T (U, W )) = 0. On the other hand since D is totally geodesic we obtain R(H, V, ξ, H) = 0 and consequently ρ(V, ξ) = 0 which means that D is an eigendistribution of the Ricci tensor ρ of ((0, L) × P, g, J). From O’Neill formulae ([12, p. 465]) it follows also that R(JH, U, V, JH) = 0 if g(U, V ) = 0 and 2 2   R(JH, U, U, JH) = s4rf4 − ff rr , for a unit vector field U as above. Note also that the distribution D spanned by the vector fields ξ, H is totally geodesic. Consequently if X, Y, Z ∈ Γ(D) and V is as above then R(X, Y, Z, V ) = 0. On the other hand for U, V as above and with g(U, V ) = 0, g(U, U ) = g(V, V ) = 1 we get (r )2 ˆ ˆ R(U, V, V, U ) = R(U, V, V, U ) − g(T (V, V ), T (U, U )) = R(U, V, V, U ) − 2 r and consequently ρ(V, V ) = ρˆ(V, V ) − (2n − 3)

(r )2 s2 f 2 f  r )g(V, V ) g(V, V ) + ( − r2 4r4 fr

for any V ∈ E. It means that at any point the number of eigenvalues of ρ is k + 1 where k is the number of eigenvalues of the Ricci tensor of (N, h) at the corresponding point. Hence in general the special Kähler potential is not the special Kähler–Ricci potential. However the distributions D, E are still orthogonal with respect to ρ. Theorem 9.1. Let F be a holomorphic, complex, homothetic foliation by curves on a simply connected, compact Kähler manifold (M, g, J). Let us assume that the form α does not vanish identically on M and the leaves of F are totally geodesic. Then M = P(L ⊕ O) where p : L → N is a holomorphic line bundle over compact, simply connected Hodge manifold (N, h), whose curvature form equals Ω = sω h , s = 0 and with a metric defined on the dense open subset M  = (0, L) × P ⊂ M g = dt2 + (

2rr 2 2 ) θ + r(t)2 p∗ h, s

where r satisfies the boundary condition described in section 7. The leaves of F are the fibers CP1 of the bundle p : P(L ⊕ O) → N . If α = 0 then M = CP1 × N where N is simply connected Kähler manifold with the product metric and the leaves of F are CP1 × {y0 } where y0 ∈ N . Proof. In view of Theorem 6.3 the distribution D associated with the foliation F coincides in an open dense subset U of M with D = span{∇τ, J∇τ }, where τ is a special Kähler potential. If dim M = 4 the potential τ is a special Kähler–Ricci potential. Using the results in section 4 we can apply the methods and proofs from [5,6]. These proofs are also valid if we assume only that τ is a special Kähler potential instead a special Kähler–Ricci potential and apply the results from section 4. Hence we do not assume that (N, h) is an Einstein manifold if dim M ≥ 6. Note that the function τ has two critical submanifolds N, N ∗ of complex co-dimension 1, since otherwise F defined in M  = M − (N ∪ N ∗ ) would not extend to the foliation on s h the whole of M . In our case s = 0 where the curvature form of the bundle L is Ω = sω h . Hence 2π ω is an integral form and (N, h) is a Hodge manifold. In the notation of [5,6] our notation can be translated as 2 follows. Let us denote u = 2s ξ and Q = s4 f 2 = r2 (r )2 . We also have V = D, H = E. Then u is a Killing vector field u = J(∇

r2 ) 2

W. Jelonek / Differential Geometry and its Applications 46 (2016) 119–131

131

2

and τ = r2 , c = 0, a = 2s . The dilation of F is 1r . We have JH = f1 ξ and ξ = 1s J(∇r2 ). It is easy to see that ∇H ξ = f  JH and if X ∈ V, then ∇X ξ = 2s ΛJX = f  JX, where Λ = 2s f  is an eigenvalue of s s c H τ corresponding to the eigendistribution V. We also have θ = 2Q g(u, ) = 2Q d τ and Θ = (r )2 is the eigenvalue corresponding to the eigendistribution H. The distance between N and N ∗ is τmax

L=

dτ √ . Q

τmin 



Note that divE H = 2(n − 1) rr and divE ξ = 0. In particular κ = 2(n − 1) rr = 0 on an open and dense subset  and |α| = 2 rr , α = 2d ln r. On the other hand for every Hodge manifold (N, h) we can construct on the manifold P(L ⊕ O) many Kähler metrics g in such a way that fibers of the bundle p : P(L ⊕ O) → N form a totally geodesic, holomorphic complex homothetic foliation. If F is a holomorphic complex homothetic foliation by curves on a simply connected complete Kähler manifold (M, g, J) with α = 0, then M is a product of Riemannian surface Σ and a Kähler manifold N . This follows easily from de Rham theorem and the fact that in this case both distributions D, E are totally geodesic. If M is compact simply connected then clearly Σ = CP1 . 2 Acknowledgements The author would like to thank the referee for his/her useful suggestions. The paper was supported by Narodowe Centrum Nauki grant DEC-2011/01/B/ST1/02643. References [1] L. Bérard Bergery, Sur de nouvelles variétés riemanniennes d’Einstein, Publ. Inst. Cartan (Nancy) 4 (1982) 1–60. [2] P. Baird, J. Wood, Harmonic Morphisms Between Riemannian Manifolds, London Mathematical Society Monographs, vol. 29, Oxford University Press, Oxford, 2003. [3] S.G. Chiossi, P.-A. Nagy, Complex homothetic foliations on Kähler manifolds, Bull. Lond. Math. Soc. 44 (2012) 113–124. [4] A. Derdziński, Killing potentials with geodesic gradients on Kähler surfaces, Indiana Univ. Math. J. 61 (4) (2012) 1643–1666, http://dx.doi.org/10.1512/iumj.2012.61.4687. [5] A. Derdziński, G. Maschler, Special Kähler–Ricci potentials on compact Kähler manifolds, J. Reine Angew. Math. 593 (2006) 73–116. [6] A. Derdziński, G. Maschler, Local classification of conformally-Einstein Kähler metrics in higher dimensions, Proc. Lond. Math. Soc. (3) 87 (3) (2003) 779–819. [7] G. Ganchev, V. Mihova, Kähler manifolds of quasi-constant holomorphic sectional curvatures, Cent. Eur. J. Math. 6 (1) (2008) 43–75. [8] Andrew D. Hwang, Michael A. Singer, A momentum construction for circle invariant Kähler metrics, Trans. Am. Math. Soc. 354 (2002) 2285–2325. [9] W. Jelonek, Kähler manifolds with quasi-constant holomorphic curvature, Ann. Glob. Anal. Geom. 36 (2009) 143–159. [10] W. Jelonek, Compact conformally Kähler Einstein–Weyl manifolds, Ann. Glob. Anal. Geom. 43 (2013) 19–29. [11] S. Kobayashi, Principal fibre bundles with the 1-dimensional toroidal group, Tôhoku Math. J. 8 (1956) 29–45. [12] B. O’Neill, The fundamental equations of a submersion, Mich. Math. J. 13 (1966) 459–469. [13] H. Pedersen, A. Swann, Riemannian submersions, four manifolds and Einstein–Weyl geometry, Proc. Lond. Math. Soc. 66 (1993) 381–399. [14] P. Sentenac, Construction d’une métrique d’Einstein sur la somme de deux projectifs complexes de dimension 2, in: Géométrie riemannienne en dimension 4 (Séminaire Arthur Besse 1978–1979), Cedic-Fernand Nathan, Paris, 1981, pp. 292–307. [15] M. Svensson, Harmonic morphisms in Hermitian geometry, J. Reine Angew. Math. 575 (2004) 45–68. [16] I. Vaisman, Conformal foliations, Kodai Math. J. 2 (1979) 26–37. [17] J.C. Wood, Harmonic morphisms and Hermitian structures on Einstein 4-manifolds, Int. J. Math. 3 (1992) 415–439.