Examples of Einstein manifolds in odd dimensions
Abstract.
We construct Einstein metrics of nonpositive scalar curvature on certain solid torus bundles over a Fano KählerEinstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricciflat metrics have slowerthanEuclidean volume growth and quadratic curvature decay. Also we construct positive Einstein metrics on certain sphere bundles over a Fano KählerEinstein manifold. We classify the homeomorphism and diffeomorphism types of the total spaces when the base is the complex projective plane.
Key words and phrases:
Conformally compact Einstein manifold, curvature, KreckStolz invariant1991 Mathematics Subject Classification:
53C251. Introduction
An Einstein manifold is a (pseudo)Riemannian manifold whose Ricci tensor is proportional to the metric tensor [4], i.e.,
(1.1) 
From the analytic point of view, the Einstein equation (1.1) is a complicated nonlinear system of partial differential equations, and it is hard to prove the existence of Einstein metrics on an arbitrary manifold. For example, it is still unknown whether every closed manifold of dimension greater than carries at least one Einstein metric [4, 0.21]. Thus people turn to studying Einstein manifolds with large isometry group, e.g., when the isometry group acts on the manifold transitively, and hence (1.1) reduces to a system of algebraic equations, or when the isometry group acts on the manifold with principal orbits of codimension one, and hence (1.1) reduces to a system of ordinary differential equations. In both cases, it becomes much more manageable to establish some existence results for Einstein metrics (see the surveys [4, Chap 7] and [30, §§2,4]). Recent progress in this direction includes the variational approach to study homogenous Einstein metrics by several authors (see [21] in the noncompact case, and [5] in the compact case).
Another natural simplification of (1.1) is to impose the Einstein condition on the total space of a Riemannian submersion with totally geodesic fibers [4, 9.61]. One major obstacle in this setup arises from the existence of YangMills connections with curvature form of constant norm. In general, it is still unknown when this necessary condition is satisfied. However, if the structure group of the underlying fiber bundle is a compact torus and the base is closed, then the curvature form of a principal connection is the pullback of a harmonic form on the base. This observation leads people to consider principal torus bundles over products of KählerEinstein manifolds and their associated fiber bundles. In these cases, the harmonic forms are rational linear combinations of the Ricci forms. Many interesting Einstein metrics have been discovered on these spaces, among which is the first inhomogeneous Einstein metric with positive scalar curvature constructed by Page [26] on , i.e., the nontrivial sphere bundle over . Later on, the method of Page was extended by several authors to construct positive HermitianEinstein metrics on certain sphere bundles over products of Fano KählerEinstein manifolds [3, 27, 29]. On the other hand, Koiso and Sakane [16] were able to construct the first inhomogeneous Fano KählerEinstein metrics on certain sphere bundles over two copies of the same Fano KählerEinstein manifolds. Note that all the aforementioned Einstein metrics live on the sphere bundles associated with a principal circle bundle. We refer the reader to [4, Chap 9] and [30, §§1,3] for more details.
In this paper, we focus on constructing smooth Einstein metrics on the solid torus bundles and the sphere bundles associated with a principal torus bundle over a single Fano KählerEinstein manifold. Note that our principal torus bundles are products of with a principal circle bundle (see Proposition 2.1).
First we consider the associated solid torus bundles. In general, given a principal torus bundle , we can construct a solid torus bundle via the left action of torus on solid torus
Our first result demonstrates the existence of conformally compact Einstein (CCE) metrics on .
Theorem 1.1.
Let be a Fano KählerEinstein manifold with first Chern class , where , and is an indivisible class. For with , let be the principal torus bundle over with characteristic classes . Then there exist a twoparameter family of CCE metrics on .
Remark 1.2.
In the case of and , we constructed in [8, Theorem 1.4] a oneparameter family of CCE metrics on .
Remark 1.3.
All the CCE manifolds Theorem 1.1 yields are of odd dimensions, among which the lowestdimensional ones are some nontrivial solid torus bundles over .
Remark 1.4.
A CCE manifold is a complete noncompact Einstein manifold which is conformal to the interior of a compact Riemannian manifoldwithboundary [13, 1, 8]. The conformal boundary of the bulk space is called the conformal infinity. The fundamental link between the global geometry of a CCE manifold and the conformal geometry of its conformal infinity lies in the asymptotic expansion of its volume function w.r.t. the geodesic defining function determined by a choice of representative for the conformal infinity. In odd dimensions, the expansion has a logarithmic term whose coefficient turns out to be independent of the choice of representative, and is identified with a constant multiple of the total curvature of the conformal infinity [14, 11]. Moreover, the pointwise curvature can be read off from the asymptotic behavior of a certain formal solution to a Poisson equation on the CCE manifold [11, Theorem 4.1]. Applying these general results to the CCE manifolds in Theorem 1.1 reveals that
Theorem 1.5.
The conformal infinity of every CCE manifold in Theorem 1.1 has a flat representative, i.e., a metric with constant zero curvature.
Remark 1.6.
Remark 1.7.
Our second result demonstrates the existence of complete Ricciflat metrics on .
Theorem 1.8.
Let be a Fano KählerEinstein manifold with first Chern class , where , and is an indivisible class. For with , let be the principal torus bundle over with characteristic classes . Then there exist a twoparameter family of complete Ricciflat metrics on .
By the Bishop volume comparison theorem, a complete noncompact Ricciflat manifold can have at most Euclidean volume growth. It can be shown that all the Ricciflat manifolds in Theorem 1.8 have slowerthanEuclidean volume growth and quadratic curvature decay (cf. [22]).
Now we turn to the associated sphere bundles. In general, given a principal torus bundle , we can construct a sphere bundle via the left action of torus on unit sphere
Our third result demonstrates the existence of positive Einstein metrics on .
Theorem 1.9.
Let be a Fano KählerEinstein manifold with first Chern class , where , and is an indivisible class. For with , let be the principal torus bundle over with characteristic classes . Then there exists a smooth positive Einstein metric on .
Remark 1.10.
In the case of and , Lü, Page and Pope [23, §3.3.2] constructed a smooth positive Einstein metric on .
Remark 1.11.
Remark 1.12.
Let be a primitive th root of unity, and let be an integer coprime to . The cyclic group of order , generated by , acts freely on from the right
It is clear from our construction that acts by isometry on the Einstein manifolds in Theorem 1.9. The quotient manifolds, which are lens space bundles, inherit positive Einstein metrics.
It is an interesting but difficult problem to study the moduli spaces of Einstein structures on the above sphere bundles. A first step in this direction is a diffeomorphism classification of the total spaces of these sphere bundles. By comparing the Einstein constants of Einstein metrics with unit volume on diffeomorphic total spaces, we may gain some information about the number of components of the moduli space .
The simplest case is that the base is . There are only two sphere bundles over up to diffeomorphism [28, §26]. The total space of the trivial one is the product manifold which is spin, while the total space of the nontrivial one is the twisted manifold which is nonspin. In Theorem 1.9, when mod , the total space is spin, and hence is diffeomorphic to , while when mod , the total space is nonspin, and hence is diffeomorphic to . It is wellknown that the moduli space of Einstein structures on either manifold has infinitely many components [12, §2.2].
The situation becomes much more involved when the base has higher dimensions. We succeed in classifying the total spaces only when is . In this case, the total space of the associated sphere bundle is a simplyconnected, closed manifold with , , and generated by , where is a generator of . So is a necessary condition for to be homotopic to . Furthermore, we can apply the classification result of Kreck and Stolz [17, 18] to show that
Theorem 1.13.
Assume and .

When , is homeomorphic (diffeomorphic) to iff .

When , is homeomorphic (diffeomorphic) to iff mod (and mod ), where
Remark 1.14.
is not homotopic to any AloffWallach space . The argument goes as follows. Note that is spin, and is a finite cyclic group of odd order. For to be spin, has to be odd (see Lemma 6.26). So is even, and is of even order. However, we do not know whether can be homeomorphic to any Eschenburg space. Note that the KreckStolz invariants for a certain type of Eschenburg spaces can be found in [20].
Theorem 1.13 provides infinitely many pairs of homeomorphic manifolds which are not diffeomorphic, as well as infinitely many pairs of diffeomorphic manifolds. For instance,
Example 1.15 (Spin case).
For , and are homeomorphic. They are diffeomorphic iff mod .
Example 1.16 (Nonspin case).
For , and are homeomorphic. They are diffeomorphic iff with mod .
Remark 1.17.
Given a pair of diffeomorphic manifolds as in Examples 1.15 and 1.16, Theorem 1.9 asserts the existence of two positive Einstein metrics with unit volume on the underlying smooth manifold. But it is hard to verify whether they have the same Einstein constants or not (cf. Remark 6.23). We believe that the Einstein constants should be different in general. Thus the moduli space of Einstein structures would have more than one component.
The remainder of this paper is structured as follows. In §2, we discuss a class of principal torus bundles over a Fano KählerEinstein manifold from both topological and geometric viewpoints. In §3, we compute Ricci curvatures of the warped product of an open interval and a principal torus bundle studied in §2, and reduce the Einstein equation on the product to a system of ODEs. In §4, we find exact solutions to a subsystem of the Einstein system derived in §3. In §5, we construct complete nonpositive Einstein metrics on associated solid torus bundles. After that comes a proof of Theorem 1.5. In §6, we construct positive Einstein metrics on associated sphere bundles. We end the paper with a detailed calculation of characteristic classes of the sphere bundles and the ball bundles they bound, which leads to an argument for Theorem 1.13.
2. Principal torus bundles
This section is devoted to a brief discussion of the topological and geometric properties of principal torus bundles involved in Theorems 1.1 and 1.9 (see [31] for more details).
2.1. Topology
Let be a Fano KählerEinstein manifold of complex dimension . Assume that the first Chern class , where is the Einstein constant of , i.e., , and is an indivisible class. Notice that is torsionfree as is simplyconnected.
Let be dimensional compact torus. We decompose once and for all as a product , and choose a basis for its Lie algebra . Thus the set of isomorphism classes of principal bundles over is identified with , i.e., the set of homotopy classes of maps from to . Since is the EilenbergMacLane space , is isomorphic to . Therefore a principal bundle over is classified by a pair of characteristic classes .
The following describes how to construct the principal bundle classified by a given pair of characteristic classes. For , let be the principal bundle over with Euler class . Their Cartesian product is a principal bundle over . The pullback bundle of by the diagonal map is the principal bundle over with characteristic classes .
Observe that has a large automorphism group . This fact provides a simple way to construct new principal bundles out of old ones as follows. Given a principal bundle with characteristic classes , we can change the action on via an element , say .
This yields a new principal bundle with characteristic classes , where . Notice that and have the same manifold as their total spaces.
From now on, we will focus on a special class of principal bundles over , whose characteristic classes are both integral multiples of . Given , let be the principal bundle over with characteristic classes . Denote by the greatest common divisor of and , and let be the principal bundle over with Euler class . We have
Proposition 2.1.
The total spaces of and are diffeomorphic to each other.
Proof.
Bézout’s identity asserts the existence of integers and such that . Let
Changing the action on via gives rise to a new principal bundle over with characteristic classes . But is also a principal bundle over with characteristic classes . Thus is diffeomorphic to . The proposition follows from the fact that and have the same total space.
2.2. Geometry
Let be the principal bundle with Euler class , . There is a principal connection on with curvature form , where is the Kähler form associated with . If we denote by the global vertical vector field on generated by , i.e.,
then .
Recall that the principal bundle is the pullback bundle of the Cartesian product via the diagonal map . Thus a principal connection on is given by the pullback of , and its curvature form is .
Let be an arbitrary positivedefinite symmetric matrix. It induces a leftinvariant metric on given by . Consider now a bundle metric on
(2.1) 
where is a positive constant, and the convention is . Such a choice makes into a Riemannian submersion with totally geodesic fibers. More importantly, the curvature form is parallel w.r.t. the base metric. Thus the principal connection satisfies the YangMills condition. The relationship of the O’Neill tensor field (cf. [4, 9.20]) and the curvature form becomes
where and are arbitrary horizontal vector fields on .
Now we compute the Ricci tensor of . To do that we choose an orthonormal adapted basis on , i.e., , , and let be the unique horizontal vector field on such that . This gives us a standard basis on . In terms of such a basis, we have [4, 9.36]
Lemma 2.2.
The nonvanishing components of the Ricci tensor of are
with and .
Remark 2.3.
since is positivedefinite, and .
Let be the Ricci endomorphism of , i.e., , for . By Lemma 2.2, we have
Lemma 2.4.
where and are orthonormal vertical vector fields, and . The scalar curvature of is
Finally we derive a useful formula for involving , and . Observe that the dual coframe of consists of and , i.e., . In terms of , the vertical components of the metric (2.1) takes a diagonal form
Comparing the coefficients on both sides yields
Lemma 2.5.
3. A family of principal bundles
Let be a principal bundle as in §2, and let and be respectively the associated solid torus bundle and sphere bundle as in §1. Notice that there exist diffeomorphisms
where is an open interval. Thus our strategy for proving Theorems 1.1, 1.8 and 1.9 consists of constructing local metrics on and , and then extending them to be smooth Einstein metrics on and respectively.
In this section, we study the geometry of product endowed with metric , where
(3.1) 
is a smooth oneparameter family of positivedefinite symmetric matrices, and is a smooth positive function of . We will reduce the Einstein equation (1.1) on to a system of ODEs.
Denote by the selfadjoint shape operator of hypersurfaces . By definition,
where is the LeviCivita connection of , and is the unit normal vector.
Lemma 3.1.
For , we have
where denotes .
Proof.
Given a vector field on , we denote by its unique independent extension to , such that the Lie bracket , i.e., . Thus
Lemma 3.2.
On the standard basis , can be written in a matrix form
where , is the inverse matrix of , and is the identity matrix. In particular,
Proof.
Corollary 3.3.
On the standard basis , can be written in a matrix form
In particular,
Now we compute the Ricci curvatures of . For vector fields , we have
where is the LeviCivita connection of . Thus
Lemma 3.4.
, .
Proof.
Notice that
where we have used the fact since the fiber of the Riemannian submersion is an abelian group. But is vertical, so is . Thus .
Claim 3.5.
.
Proof.
In fact,
It follows from Claim 3.5 that
Similarly, we have . Moreover,
where is the O’Neill tensor field associated with (cf. [4, 9.20]).
To sum up, we see that .
Lemma 3.6.
, .
Proof.
We have