Group actions on manifolds of positive sectional curvature

Burkhard Wilking University of Pennsylvania 209S. 33rd Street, Philadelphia PA 19104

Mathematische Arbeitstagung June 13-19, 2001 Max-Planck-Institut für Mathematik, Bonn, Germany

There are only very few examples of compact Riemannian manifolds of positive sectional curvature known. In dimensions above 24 all known simply connected examples are diffeomorphic to rank 1 symmetric spaces, 𝕊^{n}, ℂℙ^{n} or ℍℙ^{n}. On the other hand there are only very few constrains known. For example there is not a single obstruction known that distinguishes the class of simply connected compact manifolds that admit positively curved metrics from the class of simply connected compact manifolds that admit non-negatively curved metrics, although the class of known examples is much bigger in the latter case.

Karsten Grove [1991] proposed to study isometric group actions on manifolds of positive sectional curvature and to find obstructions for such actions. One of the hopes is that a systematic study of the problem might give ideas how one can construct new examples.

The main aim of the talk is to present a few new results for isometric actions on manifolds of positive sectional curvature. One of the main new tools is

Theorem 1 Let (M^{n}, g) be a closed oriented Riemannian manifold with positive sectional curvature, and let N^{n−k} be a totally geodesic compact oriented submanifold without boundary. Suppose that the codimension k satisfies k ≤ \frac{n}{3}. Then there is a cohomology class e ∈ H^{k}(M, ℤ) such that

∪e: H^{i}(M, ℤ) → H^{i+k}(M, ℤ)

is an isomorphism for k ≤ i ≤ n − 2k.

Given a Riemannian manifold with positive sectional curvature and with a ’large’ amount of symmetry we can find totally geodesic submanifolds of low codimension by considering fixed point sets of isometries. By Theorem 1 this gives a period in the cohomology ring. This is a vital part in the proofs of the following results.

In order to measure the amount of symmetry of a Riemannian manifold (M^{n}, g) we consider its isometry group Iso(M, g) and assign to the pair the following three constants

symrank(M, g) := rank(Iso(M, g)),
symdeg(M, g) := dim(Iso(M, g)) and
cohom(M, g) := dim((M, g)/Iso(M, g)),

where the last constant denotes the dimension of the orbit space or equivalently the codimension of a principal orbit. Our main results are

Theorem 2 Let (M^{n}, g) be a simply connected Riemannian manifold of positive sectional curvature. Suppose that n ≥ 10 and

symrank(M, g) ≥ \{ \begin{matrix}\frac{n}{4} + 2 & if n ≡ 3 mod 4 or n = 24, 25, 26. \\ \frac{n}{4} + 1 & otherwise \end{matrix}

Then M is homotopically equivalent to 𝕊^{n}, ℂℙ^{n} or ℍℙ^{n/4}.

In dimensions 12, 13 and 16 Theorem 2 turns out to be optimal.

Theorem 3 Let (M^{n}, g) be a simply connected Riemannian manifold of positive sectional curvature. Suppose that

symdeg(M, g) ≥ 2n − 5

Then M is homotopically equivalent to a rank 1 symmetric space or isometric to a homogeneous space of positive sectional curvature.

The simply connected homogeneous spaces of positive sectional curvature have been classified by Berger [1961], Wallach [1972] and Berard Bergery [1976]. Apart from the rank 1 symmetric spaces there are exceptional spaces in the dimensions 6, 7, 12, 13 and 24 and all of these satisfy the hypothesis of Theorem 3.

Theorem 4 Let (M^{n}, g) be a simply connected Riemannian manifold of positive sectional curvature. Put k := cohom(M, g) and assume for the dimension n that n ≥ (30(k + 1))^{2}. Then M is homotopically equivalent to a rank 1 symmetric space.