Geometrically Formal Homogeneous Metrics of Positive Curvature

Abstract

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere. Compact manifolds of positive sectional curvature form an intriguing field of study. On the one hand, there are few known examples, and on the other hand the two main conjectures in the subject, the two Hopf conjectures, are still wide open. The most basic examples of positive curvature are the rank one symmetric spaces S, CP, HP and CaP. Homogeneous spaces of positive curvature have been classified [Be, BB]: there are the homogeneous flag manifolds due to Wallach, W 6 = SU(3)/T , W 12 = Sp(3)/ Sp(1) and W 24 = F4/ Spin(8), the Berger spaces B 7 = SO(5)/ SO(3) and B = SU(5)/ Sp(2) · S, and the Aloff–Wallach spaces W 7 p,q = SU(3)/ diag(z, z, z̄) with gcd(p, q) = 1, p ≥ q > 0. See e.g. [Zi2] for a detailed discussion. Furthermore, we have the biquotient examples due to Eschenburg [E1, E2] and Bazaikin [Baz] and the more recent cohomogeneity one example in [De, GVZ]. All the known examples have the following remarkable properties: They are rationally elliptic spaces, i.e. their rational homotopy groups πi(M)⊗Q vanish from a certain degree i on, and the even dimensional ones have positive Euler characteristic. For general simplyconnected positively (or more generally non-negatively) curved manifolds, the Bott-GroveHalperin conjecture claims rational ellipticity, whilst the Hopf conjecture asserts that their Euler characteristic is positive in even dimensions. A (simply-connected) topological space is called (topologically) formal if its rational homotopy type is a formal consequence of its rational cohomology algebra, or, equivalently in the case of a manifold, if its real cohomology algebra is weakly equivalent to its de Rham algebra. It is a classical result of rational homotopy theory that rationally elliptic spaces with positive Euler characteristic are formal, see e.g. [FHT]. In fact, one easily sees that all known examples of positive curvature are formal, in even as well as in odd dimensions. It is thus natural to conjecture that positively curved manifolds are formal in general. We mention here that the situation is different in non-negative curvature. Homogeneous spaces G/H naturally admit non-negative curvature and are rationally elliptic. If rkH = The first author was supported by IMPA and a research grant of the German Research Foundation DFG. The second author was supported by CAPES-Brazil, IMPA, the National Science Foundation and the Max Planck Institute in Bonn. 1 2 MANUEL AMANN AND WOLFGANG ZILLER rkG they have positive Euler characteristic and are hence formal. On the other hand, in [Am, KT3] one finds many examples of non-formal homogeneous spaces. Other classical examples of formal spaces are compact symmetric spaces and compact Kähler manifolds. In the case of symmetric spaces this simply follows from the fact that harmonic forms are parallel. Thus in [Ko1] the notion of geometric formality was introduced: A Riemannian metric is geometrically formal if wedge products of harmonic forms are again harmonic. On a compact manifold the Hodge decomposition implies that a manifold admitting a geometrically formal metric is also topologically formal. See [Ba] and [Ko2] for some recent results on geometrically formal metrics in dimension 3 and 4, and [Ko1, KT1, KT2, KT3, OP, GN] for obstructions to geometric formality. There are very few known examples of compact geometrically formal manifolds. In fact, to our knowledge they all belong to the following classes (see [Ko1, Ko2, KT3, Ba]) • a Riemannian metric all of whose harmonic forms are parallel, • a homogeneous metric on a manifold whose rational cohomology is isomorphic to the cohomology of S×S with either p and q both odd, or p even and q odd with p > q, • Riemannian products of the above and finite quotients by a group of isometries. In the homogeneous case geometric formality is an obvious consequence of homogeneity, since harmonic forms must be invariant under the id component of the isometry group. Homogeneous spaces which have the rational cohomology of the product of spheres are classified in [Kr], and in [KT3] it was shown that many of them are not homotopy equivalent to symmetric spaces. There are other metrics where all harmonic forms are parallel, besides the compact symmetric spaces. For example, any metric on a rational homology sphere or a Kähler metric on a rational CP, e.g. the twistor space of the quaternionic symmetric space G2 / SO(4). If one allows the manifold not to be simply connected, there are many such examples, e.g. fake CP and CP, see [PY], which are compact quotients of complex hyperbolic space. Although these spaces may be called topologically formal, this property usually has not the strong consequences known from rational homotopy theory unless the space is nilpotent. For quotients of products, as for example (M × R)/Γ with M geometrically formal, one simply observes that geometric formality is a local property. It is the main result of this article that geometric formality is also rare in positive curvature: Theorem. A homogeneous geometrically formal metric of positive curvature is either symmetric or a metric on a rational homology sphere. In [KT3],Theorem 25, it was shown that a metric on a non-trivial S bundle over CP cannot be formal. This includes the 6 dimensional flag manifold W , as well as the inhomogenous Eschenburg biquotient. We will show that any homogeneous metric on the other two flag manifolds W 12 and W 24 cannot be geometrically formal. Of course, every metric on a sphere is geometrically formal, and every homogeneous metric on CP, HP and CaP is symmetric. The Berger space B is geometrically formal as well, since it is a rational homology sphere. This leaves the Berger space B, the Aloff–Wallach spaces, and the homogeneous metrics on CP. For the Aloff–Wallach spaces, it was shown in [KT3] that GEOMETRICALLY FORMAL HOMOGENEOUS METRICS OF POSITIVE CURVATURE 3 the normal homogeneous metric is not geometrically formal, but this metric does not have positive curvature. The recent example of positive curvature in [De, GVZ] is a rational homology sphere and hence geometrically formal. It would be interesting to know if the only other known examples of positive curvature, i.e. the 7 dimensional Eschenburg spaces and 13 dimensional Bazaikin spaces, can admit geometrically formal metrics. They have the same cohomology as Wp,q and B , but our methods do not apply in this case since the isometry group is too small. It would also be interesting to have some other examples of homogeneous spaces where some of the homogeneous metrics are geometrically formal. Although the methods in this paper can be used to check this, an example seems to be difficult to find. Any relationship in the cohomology ring puts strong restrictions on a geometrically formal metric. To prove the theorem we use the elementary fact that the de Rham cohomology is isomorphic to the finite dimensional algebra of invariant forms, and hence closed and harmonic forms can be computed explicitly. The Berger space B has the rational cohomology of CP × S and the Aloff–Wallach space Wp,q that of S × S. Hence there is a unique harmonic 2-form η and to be geometrically formal implies that η resp. η must be 0 as a form. It turns out that even among the closed invariant forms there are none whose power is 0. For W 12 and W 24 there are relations in the cohomology ring that contradict geometric formality. In the case of CP, the situation is more interesting. Here the condition is that η must be harmonic for all k. But already the harmonic 4-form changes with the metric and is the square of the harmonic 2-form only if the metric is symmetric. We point out that this metric is also almost Kähler, hence gives examples of such metrics which are not geometrically formal. In Section 1 we explain some background about homogeneous spaces and their cohomology. In Section 2 we deal with B and in Section 3 with the Aloff–Wallach spaces. In Section 4 we discuss W 12 and W , and in Section 5 CP.

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Cite this paper

@inproceedings{Amann2014GeometricallyFH, title={Geometrically Formal Homogeneous Metrics of Positive Curvature}, author={Manuel Amann and Wolfgang Ziller}, year={2014} }