On Weyl-reducible conformal manifolds and lcK structures

TL;DR

This paper explores the structure of Weyl-reducible conformal manifolds, showing restrictions on their universal covers and characterizing cases related to Anosov diffeomorphisms, thus advancing understanding of their geometric and dynamical properties.

Contribution

It extends Kourganoff's results by establishing that the parameter q must be 1 or 2 in certain Weyl-reducible conformal manifolds, refining the classification of these structures.

Findings

Universal cover splits as R^q times an incomplete manifold N

If dim(N)=2, then q=1 or 2

Provides constraints on the structure of Weyl-reducible conformal manifolds

Abstract

A recent result of M. Kourganoff states that if $D$ is a closed, reducible, non-flat, Weyl connection on a compact conformal manifold $M$, then the universal covering of $M$, endowed with the metric whose Levi-Civita covariant derivative is the pull-back of $D$, is isometric to $\mathbb{R}^q\times N$ for some irreducible, incomplete Riemannian manifold $N$. Moreover, he characterized the case where the dimension of $N$ is $2$ by showing that $M$ is then a mapping torus of some Anosov diffeomorphism of $T^{q+1}$. We show that in this case one necessarily has $q=1$ or $q=2$.

Equations

$${\theta }_{g^{\prime }}={\theta }_g+\mathrm{d}f.$$

$$(\tilde{M},g_D)={\mathbb{R}}^q\times (N,g_N).$$

$$g_{\phi }:=\mathrm{d}{x}_1^2+\cdots +\mathrm{d}{x}_q^2+\phi (t)\mathrm{d}{x}_{q+1}^2+\mathrm{d}{t}^2,$$

$$\lambda A{|}_{E^s}\in \mathrm{O}(E^s,b)⟺C\cdot (\lambda A{|}_{E^s})\cdot C^{-1}\in \mathrm{O}(q).$$

$${\mu }_A(X)=(X-{\lambda }^q)\prod _{j=1}^q\left(X-\frac{z_j}{\lambda }\right),$$

$${\lambda }^{2p}+\frac{1}{\lambda }\sum _{j=1}^{2p}{z}_j=-a_{2p},{\lambda }^{-2p}+\lambda \sum _{j=1}^{2p}\frac{1}{z_j}=a_1.$$

$${\lambda }^{4p+2}+a_{2p}{\lambda }^{2p+2}+a_1{\lambda }^{2p}-1=0.$$

$$Q(X):=X^{2p+1}+a_{2p}{X}^{p+1}+a_1{X}^p-1.$$

$${\lambda }^2+\sum _{j=1}^{2p}{r}_j=0=\frac{1}{{\lambda }^2}+\sum _{j=1}^{2p}\frac{1}{r_j}.$$

$$-{\lambda }^{-2}=\sum _{j=1}^{2p}\frac{1}{r_j}={\lambda }^{\frac{2}{p}}\sum _{j=1}^{2p}{r}_j=-{\lambda }^{\frac{2}{p}}{\lambda }^2.$$

$$\mathrm{d}\Omega =\theta \wedge \Omega ,$$

$$F:\text{doba}C\times \text{doba}{H}^s\to \text{doba}R,F(z,z_1,\dots ,z_s):=|z{|}^2+\frac{1}{y_1\dots y_s},$$

$$u:\text{doba}{H}^s\to \text{doba}R,u(z_1,\dots ,z_s):=\frac{1}{y_1\dots y_s}=\frac{(2i{)}^s}{{\prod }_{j=1}^s(z_j-{\bar{z}}_j)},$$

$$\bar{\partial }u=u\sum _{j=1}^s\frac{\mathrm{d}{\bar{z}}_j}{z_j-{\bar{z}}_j},\partial u=-u\sum _{j=1}^s\frac{\mathrm{d}{z}_j}{z_j-{\bar{z}}_j},$$

$$\partial \bar{\partial }u$$

$$\partial \bar{\partial }u=\frac{u}{4}\sum _{j,k=1}^s\frac{1+{\delta }_{jk}}{y_j{y}_k}\mathrm{d}{z}_j\wedge \mathrm{d}{\bar{z}}_k.$$

$$(h_{j\bar{k}})=\frac{u}{4}\left(\begin{array}{cccc}\frac{1}{y_1}& 0& \dots & 0\\ 0& \frac{1}{y_2}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & \frac{1}{y_s}\end{array}\right)\left(\begin{array}{cccc}2& 1& \dots & 1\\ 1& 2& \dots & 1\\ ⋮& ⋮& \ddots & ⋮\\ 1& 1& \dots & 2\end{array}\right)\left(\begin{array}{cccc}\frac{1}{y_1}& 0& \dots & 0\\ 0& \frac{1}{y_2}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & \frac{1}{y_s}\end{array}\right),$$

$$\det (h_{j\bar{k}})={\left(\frac{u}{4}\right)}^s(s+1)\frac{1}{(y_1\dots y_s{)}^2}=\frac{(s+1)u^{s+2}}{4^s}.$$

$$\rho$$

Full text

1. Weyl-reducible manifolds

Let $(M,c)$ be a compact conformal manifold. A Weyl structure on $M$ is a torsion-free linear connection $D$ preserving the conformal structure $c$, in the sense that for every Riemannian metric $g\in c$, $D_{X}g=\theta_{g}(X)g$ for some $1$-form $\theta_{g}$ on $M$ called the Lee form of $D$ with respect to $g$. If $g^{\prime}:=e^{f}g$ is another metric in the conformal class, then

[TABLE]

The Weyl structure $D$ is called closed if $\theta_{g}$ is closed for one (and thus all) metrics $g\in c$ and exact if $\theta_{g}$ is exact for all $g\in c$. From the above formula we see that if $D$ is exact, so that $\theta_{g}={\rm d}f$ for some $g\in c$, then $\theta_{e^{-f}g}=0$, thus $D$ is the Levi-Civita connection of the metric $e^{-f}g\in c$.

The manifold $(M,c,D)$ is called Weyl-reducible if the Weyl structure $D$ is reducible and non-flat.

Based on some evidence given by the Gallot theorem on Riemannian cones [4], it was conjectured in [2] that every closed, non-exact Weyl structure on a compact conformal manifold is either irreducible or flat. Matveev and Nikolayevsky [7] constructed a counterexample to the general conjecture, but later on Kourganoff proved that a weaker form of this conjecture holds:

Theorem 1**.**

(cf. [6, Thm. 1.5]). A closed non-exact Weyl structure $D$ on a compact conformal manifold $M$, is either flat or irreducible, or the universal cover $\widetilde{M}$ of $M$ together with the Riemannian metric $g_{D}$ whose Levi-Civita connection is $D$, is the Riemannian product of a complete flat space $\mathbb{R}^{q}$ and an incomplete Riemannian manifold $(N,g_{N})$ with irreducible holonomy:

[TABLE]

In [6, Example 1.6] (see also [7]), examples of closed reducible Weyl structures on compact manifolds are constructed using a linear map $A\in\mathrm{SL}_{q+1}(\doba{Z})$, such that:

1. (1)

there exists an $A$-invariant decomposition $\doba{R}^{q+1}=E^{s}\oplus E^{u}$ with $\mathrm{dim}(E^{u})=1$ and $A|_{E^{u}}=\lambda^{q}\mathrm{Id}_{E^{u}}$ for some real number $\lambda>1$; 2. (2)

there exists a positive definite symmetric bilinear form $b$ on $E^{s}$, such that $\lambda A|_{E^{s}}$ is orthogonal with respect to $b$.

Then $A$ induces a diffeomorphism (also denoted by $A$) of the torus $\mathbb{T}^{q+1}$, whose mapping torus $M_{A}:=\mathbb{T}^{q+1}\times(0,\infty)/(x,t)\sim(Ax,\frac{1}{\lambda}t)$, carries a reducible non-flat closed Weyl structure $D_{\varphi}$ obtained by projecting to $M_{A}$ the Levi-Civita connection of the metric on $\mathbb{T}^{q+1}\times(0,\infty)$ given by:

[TABLE]

where $x_{1},\dots,x_{q+1}$ are the local coordinates with respect to an orthonormal basis $(e_{1},\dots,e_{q+1})$ with $e_{1},\dots,e_{q}\in E^{s}$, $e_{q+1}\in E^{u}$, and $\varphi\colon(0,+\infty)\to(0,+\infty)$ is any smooth function satisfying $\varphi(\lambda t)=\lambda^{2q+2}\varphi(t)$ for every $t\in(0,+\infty)$.

Moreover, Kourganoff proved that these are, up to diffeomorphism, the only examples of Weyl-reducible manifolds when the incomplete factor $N$ is 2-dimensional:

Theorem 2**.**

[6, Theorem 1.7]** Assume that $D$ is a closed non-exact Weyl structure $D$ on a compact conformal manifold $M$ which is neither flat nor irreducible. If the irreducible manifold $N$ given by Theorem 1 is $2$-dimensional, then $(M,D)$ is isomorphic to one of the Riemannian manifolds $(M_{A},D_{\varphi})$.

It turns out, however, that matrices $A\in\mathrm{SL}_{q+1}(\doba{Z})$ satisfying the conditions (1) and (2) above, only exist for $q=1$ or $q=2$. This is the object of the next section.

2. A number-theoretical result

Proposition 3**.**

Let $q\in\doba{N}^{*}$ and $A\in\mathrm{SL}_{q+1}(\doba{Z})$, such that there is a direct sum decomposition $\doba{R}^{q+1}=E^{s}\oplus E^{u}$ invariant by $A$, with $\dim(E^{u})=1$. If there exists a positive definite symmetric bilinear form $b$ on $E^{s}$ and a real number $\lambda>1$, such that $\lambda A|_{E^{s}}$ is orthogonal with respect to $b$, then $q\in\{1,2\}$.

Proof.

Let $C$ be a symmetric positive definite matrix, such that $b=\langle C^{2}\cdot,\cdot\rangle$, where $\langle\cdot,\cdot\rangle$ is the standard Euclidean scalar product. Then the following equivalence holds:

[TABLE]

In particular, each eigenvalue of $\mathrm{Spec(\lambda A|_{E^{s}})}$ has modulus $1$ and the characteristic polynomial of $A$ denoted by $\mu_{A}$ is given by:

[TABLE]

where $z_{j}$ are complex numbers with $|z_{j}|=1$ for all $j\in\{1,\ldots,q\}$, and $\prod_{j=1}^{q}z_{j}=1$. Note that $\mu_{A}$ is irreducible in $\mathbb{Z}[X]$, since if it were a product of two non-constant polynomials with integer coefficients, one of them would have all roots of modulus less than $1$, which is impossible. We distinguish the following two cases:

Case 1. If $q=2p$ is even, denoting $\mu_{A}(X)=\sum_{j=0}^{2p+1}a_{j}X^{j}$ with $a_{j}\in\doba{Z}$ and $a_{2p+1}=1$, $a_{0}=-1$, we get

[TABLE]

This shows that the sum $s:=\sum_{j=1}^{2p}z_{j}$ is real, and since $|z_{j}|=1$ for all $j\in\{1,\ldots,2p\}$, $s$ is also equal to $\sum_{j=1}^{2p}\frac{1}{z_{j}}$. Eliminating $s$ from the two equations above, yields

[TABLE]

Consequently, $\lambda^{2}$ is root of the polynomial

[TABLE]

Denote by $r_{1},\ldots,r_{2p}$ the other complex roots of $Q$. Newton’s relations show that there exists a monic polynomial $\widetilde{Q}\in\doba{Z}[X]$ whose roots are $\lambda^{2p},r_{1}^{p},\ldots,r_{2p}^{p}$. The monic polynomials $\mu_{A}$ and $\widetilde{Q}\in\doba{Z}[X]$ have both degree ${2p+1}$ and $\lambda^{2p}$ is a common root. Since $\mu_{A}$ is irreducible, they must coincide, so up to a permutation, one can assume that $r_{j}^{p}=\frac{z_{j}}{\lambda}$ for all $j\in\{1,\ldots,2p\}$. This shows that $\lambda^{\frac{1}{p}}r_{j}$ are complex numbers of modulus one for all $j\in\{1,\ldots,2p\}$.

If $p\geq 2$, the coefficients of $X^{2p}$ and $X$ in the polynomial $Q$ vanish, so

[TABLE]

Thus $\sum_{j=1}^{2p}r_{j}=-\lambda^{2}$ and as $|\lambda^{\frac{1}{p}}r_{j}|=1$ for all $j$,

[TABLE]

This contradicts the fact that $\lambda>1$, showing that $p=1$ and therefore $q=2$ (see also [1, Lemma 3.5]).

Case 2. If $q$ is odd, then $\mu_{A}$ has at least one further real root, so either $\frac{1}{\lambda}$ or $-\frac{1}{\lambda}$ is a root of $\mu_{A}$. Up to reordering the subscripts one thus has $z_{1}=\pm 1$. Assume that $z_{1}=1$ (the argument for $z_{1}=-1$ is the same). The monic polynomial $P\in\doba{Z}[X]$ defined by $P(X):=X^{q+1}\mu_{A}(\frac{1}{X})$ satisfies $P(0)=1$, and its roots are $\{\lambda^{-q},\lambda,\frac{\lambda}{z_{2}},\dots,\frac{\lambda}{z_{q}}\}$.

By Newton’s identities again, there exists a monic polynomial $\widetilde{P}\in\doba{Z}[X]$ with $\widetilde{P}(0)=1$, whose roots are $\{\lambda^{-q^{2}},\lambda^{q},(\frac{\lambda}{z_{2}})^{q},\dots,(\frac{\lambda}{z_{q}})^{q}\}$.

Since the monic polynomials $\mu_{A}$ and $\widetilde{P}\in\doba{Z}[X]$ (of same degree) have $\lambda^{q}$ as common root, and $\mu_{A}$ is irreducible, they must coincide. In particular $\lambda^{-q^{2}}$ is a root of $\mu_{A}$. On the other hand every root of $\mu_{A}$ has complex modulus equal to either $\lambda^{q}$ or $\frac{1}{\lambda}$. Since $\lambda>1$, we obtain $q=1$. ∎

Remark 4**.**

As pointed out by V. Vuletescu, for odd $q$, Proposition 3 also follows from a more general result of Ferguson [3], whose proof, however, is rather involved.

3. Applications

Our main application concerns locally conformally Kähler manifolds. Recall that a Hermitian manifold $(M,g,J)$ of complex dimension $n\geq 2$ is called locally conformally Kähler (in short, lcK) if around every point in $M$ the metric $g$ can be conformally rescaled to a Kähler metric. This condition is equivalent to the existence of a closed 1-form $\theta$, such that

[TABLE]

where $\Omega:=g(J\cdot,\cdot)$ denotes the fundamental 2-form. Let now $\widetilde{M}$ be the universal cover of an lcK manifold $(M,J,g,\theta)$, endowed with the pull-back lcK structure $(\tilde{J},\tilde{g},\tilde{\theta})$. Since $\widetilde{M}$ is simply connected, $\tilde{\theta}$ is exact, i.e. $\tilde{\theta}={\rm d}\varphi$, and by the above considerations, the metric $g^{K}:=e^{-\varphi}\tilde{g}$ is Kähler.

The group $\pi_{1}(M)$ acts on $(\widetilde{M},\tilde{J},g^{K})$ by holomorphic homotheties. Furthermore, we assume that the lcK structure is strict, in the sense that $\pi_{1}(M)$ is not a subgroup of the isometry group of $(\widetilde{M},g^{K})$. In particular, the Levi-Civita connection of the Kähler metric $g^{K}$ projects to a closed, non-exact Weyl structure on ${M}$, called the standard Weyl structure. Its Lee form with respect to $g$ is exactly $\theta$.

Due to the fact that the real dimension of an lcK manifold is even, applying Proposition 3 to the special case of a compact strict lcK manifold whose standard Weyl structure is reducible, we obtain the following:

Proposition 5**.**

Let $M$ be a compact Weyl-reducible strict lcK manifold. If the irreducible factor $N$ in the splitting of the universal cover $(\widetilde{M},g^{K})$ as a Riemannian product $\doba{R}^{q}\times N$ given by Theorem 1 is $2$-dimensional, then $q=2$ and thus $M$ is an Inoue surface $S^{0}$, cf. [5].

Let us remark that if in Proposition 5 we drop the assumption on the dimension of the irreducible factor, then there are many more examples of Weyl-reducible lcK structures. They are obtained on lcK manifolds constructed by Oeljeklaus and Toma [9], for every integer $s\geq 1$, on certain compact quotients $M_{\Gamma}$ of $\doba{C}\times\doba{H}^{s}$, where $\doba{H}$ denotes the upper complex half-plane, $\Gamma$ are certain groups whose action on $\doba{C}\times\doba{H}^{s}$ is cocompact and properly discontinuous (for the precise definition of $\Gamma$ and its action see [9]). We will briefly review them here.

In order to define the lcK structure on the quotient $M_{\Gamma}$, Oeljeklaus and Toma consider the function

[TABLE]

with $z_{k}=x_{k}+iy_{k}$ and claim that it is a global Kähler potential on $\doba{C}\times\doba{H}^{s}$ (note a small sign error in [9]). To check this, we introduce

[TABLE]

and compute

[TABLE]

[TABLE]

whence

[TABLE]

This shows that $i\partial\bar{\partial}u$ is the fundamental 2-form of a Kähler metric $h$ on $\doba{H}^{s}$ whose coefficients are $h_{j\bar{k}}=\frac{u}{4}\frac{1+\delta_{jk}}{y_{j}y_{k}}$.

Proposition 6**.**

The Kähler metric on $\doba{H}^{s}$ with Kähler potential $u$ is irreducible.

Proof.

The matrix $(h_{j\bar{k}})$ can be written as the product of 3 matrices

[TABLE]

so its determinant equals

[TABLE]

The usual formula for the Ricci form $\rho$ of $h$ (cf. e.g. [8, Eq. (12.6)]) together with (1) and (2) gives

[TABLE]

This shows that the Ricci tensor of $h$ is negative definite on $\doba{H}^{s}$, so $h$ is irreducible. ∎

As a consequence of Proposition 6, the Kähler metric on $\doba{C}\times\doba{H}^{s}$ with fundamental 2-form $\Omega=i\partial\bar{\partial}F=i{\rm d}z\wedge{\rm d}\bar{z}+i\partial\bar{\partial}u$ is the product of the flat metric on $\doba{C}$ with an irreducible Kähler metric on $\doba{H}^{s}$. Therefore, the induced lcK structure on the compact quotient $M_{\Gamma}$ is Weyl-reducible, and the irreducible factor of the universal cover given by Theorem 1 is exactly $N=\doba{H}^{s}$, so it has dimension $2s$.

Acknowledgments. This work has been partially supported by the Procope Project No. 42513ZJ (France)/ 57445459 (Germany). We would like to thank Victor Vuletescu for having pointed out a small gap in the original proof of Proposition 3.