Wolfgang Ziller

Curvature and symmetry of Milnor spheres

Since Milnor’s discovery of exotic spheres [Mi], one of the most intriguing problems in Riemannian geometry has been whether there are exotic spheres with positive curvature. It is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature [Hi]. On the other hand, there are many examples of exotic spheres with positive Ricci curvature (cf. [Ch1], [He], [Po], and [Na]) and this work recently culminated in [Wr] where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive Ricci curvature. This includes all exotic spheres in dimension 7. So far, however, no example of an exotic sphere with positive sectional curvature has been found. In fact, until now, only one example of an exotic sphere with nonnegative sectional curvature was known, the so-called Gromoll-Meyer sphere [GM] in dimension 7. As one of our main results we prove:

The free loop space of globally symmetric spaces

In the study of closed geodesics the free loop space A(M)={c: S1--,M} is a natural tool. It can be made into a Hilbert manifold and the energy integral is a C ~ function on A(M) whose critical points are the closed geodesics. Therefore Morse theory has been applied to find relationships between closed geodesics and the topology of A(M). The most important application is the theorem of Gromoll-Meyer [9] proving the existence of infinitely many closed geodesics under certain weak conditions on the topology of A(M). But one can also apply this theory the other way around by computing the homology of A(M) if one knows the closed geodesics for a sufficiently nice metric. This is done here for a globally symmetric M. It is shown that the critical points in A(M) form nondegenerate critical submanifolds whose index can be computed in terms of the roots of M, and that the relative homology appearing in Morse theory survives in all of A(M), at least for Z 2 coefficients. Thus the roots of M determine H,(A(M), Z2) completely, and it is easy to show that the Betti numbers bi(A(M), Z2) are unbounded if M has rank >1. This in turn implies that for any riemannian metric on M there exist infinitely many closed geodesics by the Gromoll-Meyer theorem. In proving that the citical submanifolds are nondegenerate we also compute the Poincare map of a closed geodesic and show that it has a particularly simple form. In 1.1 we shortly describe Morse theory on A(M), in 1.2 we introduce the Poincare map of a closed geodesic, and in 1.3 we collect a few facts about globally symmetric spaces with some simple consequences for closed geodesics. In 2.1 we determine the Poincare map of a closed geodesic which can be used at the same time to show that all critical submanifolds are nondegenerate. In 2.2 the index of a closed geodesic is calculated, and in 2.3 this is interpreted in terms of the roots of M with the consequence that the index of a closed geodesic turns out to be equal to the number of conjugate points. In 3.1 we show that the relative homology survives for Z 2 coefficients which is used in 3.2 to estimate the Betti numbers bi(A(M ), Z2). The consequence for the existence of infinitely many closed geodesics is discussed.

Weakly symmetric spaces

A Riemannian manifold M is called weakly symmetric, if for any two points p and q in M, there exists an isometry f of M which interchanges p and q. An equivalent condition is that for every geodesic γ(t) in M, there exists an isometryfwhich reverses the geodesic, i. e.f(γ(t)) = γ(-t). A Riemannian symmetric space is clearly weakly symmetric. These manifolds were first studied by A.Selberg [S] who showed that for a weakly symmetric space, the algebra of isometry invariant differential operators is commutative.

Lifting group actions and nonnegative curvature

We examine the question when a group acting by cohomogeneity one on the base of a principal G-bundle can be lifted to the total space and commutes with the action by G. We answer this question completely when the base of the principle bundle is 4-dimensional.

Symmetric spaces and strongly isotropy irreducible spaces

A strongly isotropy irreducible homogeneous space is a connected effective homogeneous space L/H where H is a compact subgroup of a Lie group L such that the identity component H 0 of H acts irreducibly on T~H(L/H). If we assume instead that H, but not necessarily H 0, acts irreducibly on Tetc(L/H), then L/H is called an isotropy irreducible homogeneous space. These spaces are interesting because by Schurs Lemma they all admit an L-invariant Einstein metric, and because they include the irreducible Riemannian symmetric spaces as a sub-family. The nonsymmetric strongly isotropy irreducible homogeneous spaces were classified independently by Manturov [M1,2,3], Wolf [Wol], and later by Kr~mer [Kr], with some minor omissions in the first two references. In addition, in [Wol] a detailed study of the geometric properties of strongly isotropy irreducible spaces was carried out. The isotropy irreducible spaces which are not strongly isotropy irreducible have recently been classified in [WZ3]. In a different direction, we present in this paper a direct proof of a beautiful observation of Wall concerning a correspondence between the compact simply connected irreducible symmetric spaces on the one hand and the compact strongly isotropy irreducible quotients of the classical groups on the other hand (cf. the remarks added in proof in [Wol, pp. 147-148]). This correspondence has a number of exceptions: certain Grassmannians and the isotropy irreducible space SO(7)/G 2, which is diffeomorphic to ~pT. These exceptions appear at first sight to be even more mysterious than the correspondence. But our main result is that there is an explanation, using general principles, of this correspondence as well as all its exceptions. In particular, this means that the classification of the non-symmetric strongly isotropy irreducible quotients of the classical groups may be read off from that of the irreducible symmetric spaces. Consider a compact simply connected irreducible symmetric space G/K, where G is the identity component of the isometry group, and K is the isotropy group,

Riemannian Manifolds with Positive Sectional Curvature

Of special interest in the history of Riemannian geometry have been manifolds with positive sectional curvature. In these notes we give a survey of this subject and recent developments.

Topological obstructions to fatness

Alan Weinstein showed that certain characteristic numbers of any Riemannian submersion with totally geodesic fibers and positive vertizontal curvatures are nonzero. In this paper we explicitly compute these invariants in terms of Chern and Pontrjagin numbers of the bundle. This allows us to show that many bundles do not admit such metrics.

On the topology of positively curved Bazaikin spaces

In this note we study the topology of the positively curved Bazaikin spaces. We show that the only Bazaikin space that is homotopically equivalent to a homogeneous space is the Berger space. Moreover, we compute their Pontryagin classes and linking form to conclude that there is no pair of positively curved Bazaikin spaces which are homeomorphic, at least if the order of the sixth cohomology group with integer coefficients is less than 10

On the Geometry of Cohomogeneity One Manifolds with Positive Curvature

We discuss manifolds with positive sectimal curvature on which a group acts isometrically with one dimensional quotient. A number of the known examples have this property, but some potential families for new examples in dimension 7 arise as well. We discuss the geometry of these known examples and the connection that the candidates have with self-dual Einstein metrics.

Isotropy irreducible spaces and s-representations

Abstract A simple description of isotropy irreducible spaces G K is given when G is classical.