Examples of Einstein manifolds in odd dimensions
We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kähler-Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat metrics have slower-than-Euclidean volume growth and quadratic curvature decay. Also we construct positive Einstein metrics on certain -sphere bundles over a Fano Kähler-Einstein 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, Kreck-Stolz invariant
1991 Mathematics Subject Classification:53C25
An Einstein manifold is a (pseudo-)Riemannian manifold whose Ricci tensor is proportional to the metric tensor , i.e.,
From the analytic point of view, the Einstein equation (1.1) is a complicated non-linear 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  in the noncompact case, and  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 Yang-Mills 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 Kähler-Einstein 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  on , i.e., the nontrivial -sphere bundle over . Later on, the method of Page was extended by several authors to construct positive Hermitian-Einstein metrics on certain -sphere bundles over products of Fano Kähler-Einstein manifolds [3, 27, 29]. On the other hand, Koiso and Sakane  were able to construct the first inhomogeneous Fano Kähler-Einstein metrics on certain -sphere bundles over two copies of the same Fano Kähler-Einstein 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 Kähler-Einstein 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 .
Let be a Fano Kähler-Einstein 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 two-parameter family of CCE metrics on .
In the case of and , we constructed in [8, Theorem 1.4] a one-parameter family of CCE metrics on .
All the CCE manifolds Theorem 1.1 yields are of odd dimensions, among which the lowest-dimensional ones are some nontrivial solid torus bundles over .
A CCE manifold is a complete noncompact Einstein manifold which is conformal to the interior of a compact Riemannian manifold-with-boundary [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
The conformal infinity of every CCE manifold in Theorem 1.1 has a -flat representative, i.e., a metric with constant zero -curvature.
Our second result demonstrates the existence of complete Ricci-flat metrics on .
Let be a Fano Kähler-Einstein 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 two-parameter family of complete Ricci-flat metrics on .
By the Bishop volume comparison theorem, a complete noncompact Ricci-flat manifold can have at most Euclidean volume growth. It can be shown that all the Ricci-flat manifolds in Theorem 1.8 have slower-than-Euclidean volume growth and quadratic curvature decay (cf. ).
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 .
Let be a Fano Kähler-Einstein 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 .
In the case of and , Lü, Page and Pope [23, §3.3.2] constructed a smooth positive Einstein metric on .
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 non-spin. In Theorem 1.9, when mod , the total space is spin, and hence is diffeomorphic to , while when mod , the total space is non-spin, and hence is diffeomorphic to . It is well-known 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 simply-connected, 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
Assume and .
When , is homeomorphic (diffeomorphic) to iff .
When , is homeomorphic (diffeomorphic) to iff mod (and mod ), where
is not homotopic to any Aloff-Wallach 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 Kreck-Stolz invariants for a certain type of Eschenburg spaces can be found in .
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 (Non-spin case).
For , and are homeomorphic. They are diffeomorphic iff with mod .
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 Kähler-Einstein 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 non-positive 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
Let be a Fano Kähler-Einstein 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 torsion-free as is simply-connected.
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 Eilenberg-MacLane 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
The total spaces of and are diffeomorphic to each other.
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.
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.,
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 positive-definite symmetric matrix. It induces a left-invariant metric on given by . Consider now a bundle metric on
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 Yang-Mills 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]
The non-vanishing components of the Ricci tensor of are
with and .
since is positive-definite, and .
Let be the Ricci endomorphism of , i.e., , for . By Lemma 2.2, we have
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
3. A family of principal -bundles
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
is a smooth one-parameter family of positive-definite 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 self-adjoint shape operator of hypersurfaces . By definition,
where is the Levi-Civita connection of , and is the unit normal vector.
For , we have
where denotes .
Given a vector field on , we denote by its unique -independent extension to , such that the Lie bracket , i.e., . Thus
On the standard basis , can be written in a matrix form
where , is the inverse matrix of , and is the identity matrix. In particular,
Clearly, . We need to determine the remaining components.
On the standard basis , can be written in a matrix form
Now we compute the Ricci curvatures of . For vector fields , we have
where is the Levi-Civita connection of . Thus
where we have used the fact since the fiber of the Riemannian submersion is an abelian group. But is vertical, so is . Thus .
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 .