j. differential geometry 47 (1997) 530-??

DIFFERENTIAL TOPOLOGICAL RESTRICTIONS CURVATURE AND SYMMETRY

KARSTEN GROVE & CATHERINE SEARLE

A basic question one asks in Riemannian geometry is: how are geometric properties of a manifold reflected in its topology? An analogous question in transformation groups is: what topological restrictions are forced on a manifold by the existence of an effective action of a large group? In this work, we consider a combination of these two problems, namely:

Classify positively curved manifolds with large isometry groups. One measurement for the size of a transformation group, G × M →

M, is the dimension of its orbit space, M/G, also called the cohomogeneity of the action. This dimension is clearly constrained by the dimension of the fixed point set, M^{G}, of G in M. In fact, dim(M/G) ≥ dim(M^{G}) + 1 for any non-trivial action. In light of this we define the fixed point cohomogeneity of an action by

(0.1)  cohom fix(M, G) = dim(M/G) − dim(M^{G}) ≥ 1,

that is, as the codimension of M^{G} in M/G. Note that if M^{G} = ∅, then, by convention, cohomfix(M, G) = cohom(M, G) + 1. Thus, (M, G) has minimal fixed point cohomogeneity one, if either M is homogeneous, or G acts transitively on a normal sphere to some component of M^{G}. In the latter case we say that M is fixed point homogeneous.

Recall that simply-connected homogeneous manifolds of positive sectional curvature have been classified in Berger [?]^{1}, Aloff, Wallach [?], [?], and Berard-Bergery [?]. As one of our main results, we provide a complete classification of fixed point homogeneous manifolds of postive sectional curvature (cf. Theorem 2.8). As a special case, we obtain:

Received February 2, 1996, and, in revised form, January 29, 1997. The first author was supported in part by a grant from the NSF.

1Cf. also B. Wilking, The normal homogeneous space (SU(3) × SO(2))/U(2) has positive sectional curvature, Proc. Amer. Math. Soc., to appear.

Theorem A. Any simply-connected, fixed point homogeneous manifold of positive sectional curvature is diffeomorphic to either S^{n}, CP^{m}, HP^{k} or CaP^{2}.

Another measurement for the size of G × M → M is the dimension, dim(G) of G relative to dim(M). From a dual point of view, G is large if dim(M) is small relative to G. This viewpoint is related to representation theory. In a sense, the most basic linear representations of a compact Lie group, G, are those of lowest dimension. Thus motivated, we may also interpret the above problem in the following manner:

For a given compact Lie group, G, classify the low-dimensional positively curved manifolds, M, on which G can act (almost) effectively by isometries.

Recall that any connected, compact Lie group, G is finitely covered by a group G̃ = T^{k} × G_{1} × ... × G_{l}, where each G_{i}, i = 1, ..., l, is simple. Our classification of positively curved manifolds with maximal symmetry rank in [?] can be viewed as an answer to the above problem when G = T^{k} is abelian. In this paper, we consider the remaining building blocks, i.e., the simple Lie groups and, in particular, the classical ones.

If for each compact Lie group, G, we set

rep_{0}^{+}(G) = min{n|Gacts (almost) effectively by isometries
on some M^{n} with sec(M^{n}) > 0},

(0.2)

then another main result of this paper can be stated as (cf. Theorems 3.7, 3.9, 3.11, 3.12, 3.13):

Theorem B. Let G be a connected, compact, simple Lie group other than E_{6}, E_{7}, or E_{8}. Then

(i) rep_{0}^{+}(G) = min{dim(G/H)|H ⊂ G closed subgroup },

(ii) any positively curved (almost) G-manifold, M, with dim(M) ≤ 2rep_{0}^{+}(G) − ϵ(G) is diffeomorphic to a positively curved homogeneous manifold, where ϵ(G) is a small number depending on G.

The key reduction used in the proof of this result is that any G × M → M satisfying the assumptions of Theorem B is either homogeneous, of cohomogeneity one or of fixed point cohomogeneity one. In an analagous curvature free setting we refer to the work initiated by W.-Y. Hsiang in [?] (see also [?]).

Recall that, except for the examples due to Eschenburg [?], [?] and Bazaikin [?], all known positively curved manifolds are homogeneous (up to diffeomorphism). Thus, Theorem B provides another motivation for the systematic work initiated here. Indeed, for most, if not for all groups G, the conclusion in (ii) will almost certainly fail when dim(M) is sufficiently large. It is quite likely that methods as developed in this paper, when applied to the lowest dimensional manifolds, M, where the theorem fails, will yield enough structure on M so as to propose potentially new examples of manifolds with positive curvature. However, we will refrain from pursuing this issue here. The following are simple consequences of Theorem B.

Corollary C. Let G/H be a homogeneous space of positive curvature. If M is a positively curved manifold with dim(M) = dim(G/H) on which G acts (almost) effectively by isometries. Then M is diffeomorphic to a positively curved homogeneous manifold (which is not necessarily G/H).

In this generality, the conclusion of Corollary C fails if the symmetry group G of M is replaced by the smaller group H. However, for the rank-one symmetric spaces, we have the following result.

Corollary D. Let G/H be a compact rank-one symmetric space (CROSS). If M is a positively curved manifold on which H acts (almost) effectively by isometries, and dim(M) = dim(G/H) ≥ 16, then M is diffeomorphic to a CROSS.

In both of these corollaries, the conclusion holds for manifolds with dimension larger than dim(G/H). However, Corollary D fails in dimension 7 = dim(Sp(2)/Sp(1)), namely each Aloff-Wallach example W^{7} = SU(3)/T_{k,l} admits an Sp(1) action but is not a CROSS. All in all, one might thus be tempted to phrase the main results of this paper as follows: any potentially new example of a manifold of positive curvature must have significantly smaller symmetry group than those of the known examples.

The point of departure for our investigations is to analyse transformation groups G × M → M directly via the geometry of their orbit spaces X = M/G. These spaces form a particularly beautiful subclass of the so-called Alexandrov spaces, and our work is, to a large extent, propelled by the recent progress in this area. Of particular importance to us is the fact that if M has positive curvature, then so does X. This becomes especially restrictive if X has non-empty boundary, since in that case X is contractible by the Cheeger-Gromoll-Meyer Soul theorem adapted to Alexandrov spaces (cf. [?]). Other restrictions are obtained via Alexandrov-Toponogov type angle comparisions when applied to triangles in X with vertices at singular points.

We arrive at our results when these geometric methods, together with critical point theory for distance functions (cf. e.g. [?]), are combined with known results from Lie theory and representation theory. For general facts about representation theory, we refer the reader to [?]. All claims about dimensions of representations and inclusions among Lie groups follow easily from the theory in [?]. For facts about subgroups of exceptional Lie groups, we refer to [?]. Finally, we occasionally need to compute the normalizer of a subgroup in some specific examples. For general methods as to how to do this, we refer for example to [?], in particular to paragraph 3. We wish to thank Wu-Yi Hsiang and W. Ziller for numerous illuminating conversations in which they shared their views and expertise on the latter subjects.

1. Alexandrov geometry of orbit spaces

Throughout this paper M will denote a complete, connected Riemannian n-manifold, and G a compact Lie group which acts (almost) effectively on M by isometries. The orbit space X = M/G is equipped with the orbital distance metric from M.

Although we are primarily interested in positively curved manifolds, the natural setting for our methods applies to manifolds, M, whose sectional curvature is bounded from below, i.e., sec(M) ≥ k. It is well known that there are many geometrically equivalent formulations of the condition sec(M) ≥ k, some of which involve distances only (cf. e.g. [?], [?] and [?]). It is therefore easy to see, that, in this distance comparision sense, X is curved from below as well, in fact, curv(X) ≥ k. Thus, X is an example of a so-called Alexandrov space, and

(1.1)  dim(X) = cohom(M, G)

by definition of the cohomogeneity of the action G × M → M.

The local and infinitesimal structure of general Alexandrov spaces is tied to spaces of directions (cf. [?], [?]). In the case of orbit spaces X = M/G, these are described as follows. For p ∈ M, we denote its orbit in M by G(p), and when viewed as a point in X by p̄. The space of directions, Sp̄X at p̄ ∈ X, consists exclusively of geodesic directions. Moreover,

(1.2)  Sp̄X = S_{p}^{⊥}/G_{p},

where S_{p}^{⊥} is the unit normal sphere to G(p) at p, and G_{p} = {g ∈ G : gp = p} is the isotropy group of p. Note that p̄ is a euclidean point of X, i.e., Sp̄X = S_{1}^{m−1}, where S_{1}^{m−1} is the unit (m − 1)-sphere, m = dim(X), if and only if G(p) is a principal orbit in M. We denote the set of such points by M_{e}, and call it the regular part of M. Correspondingly, M_{s} = M − M_{e} is called the singular part of M.

As a first application of comparision theory, we show how curv(X) ≥ k imposes restrictions on the singular set M_{s}, via X_{s} = M_{s}/G. For simplicity, we confine ourselves to the case where k ≥ 0, since otherwise the diameter of X must be invoked.

Extent Lemma 1.3. For any choice of (q + 1) distinct points p̄_{0}, ...., p̄_{q} ∈ X = M/G one has

\frac{1}{q + 1}Σ_{i=0}^{q}xt_{q}Sp̄_{i}X \begin{matrix}> \\ (=) \end{matrix} \frac{π}{3}

whenever curv(X) \begin{matrix}> \\ (=) \end{matrix} 0.

Proof. Join each pair of points from {p̄_{0}, ..., p̄_{q}} by a segment in X, and add up all angles between pairs of segments with common endpoints. This is carried out in two different ways: (i) takes the sum for each triangle and then add up over all triangles; (ii) takes the sum at each point and then add up over all points. Thus

( \begin{matrix}q \\ 2 \end{matrix} ) Σ_{i=0}^{q}xt_{q}Sp̄_{i}X ≥ Σangles ≥ π ( \begin{matrix}q+1 \\ 3 \end{matrix} ) ,

where (i) and curv(X) ≥ 0 have been used for the right-hand inequality, and (ii) together with the definition of the q-extent (the maximal average distance between q points, cf. [?]) have been used in the left-hand inequality. q.e.d.

In this form, (1.3) is a powerful simple generalization of one of the key ideas applied in [?]. Note that Sp̄ and hence xt_{q}Sp̄ is smaller the more singular p̄ is. Thus (1.3) yields quantitative restrictions on the number of singular orbits of various types when sec(M) ≥ 0. For example, there can be at most two points p̄_{0}, p̄_{1} ∈ X with diamSp̄_{i} ≤ \frac{π}{3} if curv(X) > 0. In the slightly more restrictive case in which diam(Sp̄_{i}) ≤ \frac{π}{4} , M can be described as follows:

Equivariant Sphere Theorem 1.4. Let M be a closed manifold with sec(M) > 0 on which G acts (almost) effectively by isometries. Suppose p_{0}, p_{1} ∈ M are points such that diamSp̄_{i} ≤ \frac{π}{4} , i = 0, 1. Then M can be exhibited as

M = D(G(p_{o})) ⋃_{E} D(G(p_{1})),

where D(G(p_{i})), i = 0, 1 are tubular neighborhoods of the p_{i}-orbits and E = ∂D(G(p_{0})) = ∂D(G(p_{1})). In particular, M is homeomorphic to a sphere if G(p_{i}) = p_{i}, i.e., if p_{i}, i = 0, 1 are isolated fixed points for G and diamSp̄_{i} ≤ \frac{π}{4} .

Proof. Let p ∈ M − (G(p_{0}) ⋃ G(p_{1})) be chosen arbitrarily. Since curv(X) > 0, it follows from the assumption that ∠(c_{0}, c_{1}) > \frac{π}{2} for any segment c_{i} from p̄ to p̄_{i}, i = 0, 1. In M this means that p is a regular point for the distance functions, dist(G(p_{i}), ·), i = 0, 1, and the claim follows from the isotopy lemma (cf. e.g. [?]). q.e.d.

Note that only curv(X) > 0, not sec(M) > 0, is used in the proof. In particular, the structure of any closed manifold of cohomogeneity one with finite fundamental group is recovered in (1.4).

Even when the singularities of X = M/G are too mild for (1.3) to apply (e.g. when diamSp̄_{i} > \frac{π}{2} and thus π), they often yield interesting restrictions in a different way. The most remarkable one of these arises when X has non-empty boundary. Here p̄ ∈ ∂X ⊂ X_{s} by definition, if ∂Sp̄X ≠ ∅. This inductive definition is anchored by the simple fact that the only compact 1-dimensional orbit spaces (Alexandrov Spaces) are closed intervals and circles.

Now suppose that ∂X ≠ ∅ and curv(X) > 0. Then the Soul theorem adapted to Alexandrov spaces by Perelman [?] asserts that:

(1.5)  dist(∂X, ·) : X → R is strictly concave.

In particular, this tells us the following:

(1.6) there is a unique point p̄_{1} ∈ X at maximal distance from ∂X, (1.7) for any p̄ ∈ X − (∂X ⋃{p̄_{1}}) and segments c_{0}, c_{1} from p̄ to ∂X,

and p̄_{1}, respectively, one has ∠(c_{0}, c_{1}) > \frac{π}{2} ,

(1.8) X is contractible.

If π : M → M/G = X is the quotient map, we let M_{∂} ⊂ M_{s} denote the subset defined by M_{∂} = π^{−1}∂(X). Moreover, for any subset A of M (or of X), and any r > 0 we use D(A, r) to denote the closed r-neighborhood of A. Correspondingly, S(A, r) is the set of points at distance r to A and B(A, r) = D(A, r) − S(A, r).

As a fairly straightforward consequence of (1.2), (1.6) and (1.7) combined with critical point arguments for dist(M_{∂}, ·) and dist(G(p_{1}), ·) (via dist(∂X, ·) and dist(p̄_{1}, · )) (cf. e.g. [?] or [?]) one derives the following basic:

Soul Lemma 1.9. Suppose M is a closed manifold with sec(M) > 0, on which a compact Lie group G acts (almost) effectively by isometries, such that ∂(M/G) ≠ ∅. Then,

(i) there is a unique orbit, G(p_{1}) ⊂ M at maximal distance from M_{∂} ⊂ M,

(ii) for any p ∈ M − (M_{∂} ⋃ G(p_{1})), the intersections M_{∂} ⋂ M^{G_{p}} and G(p_{1}) ⋂ M^{G_{p}} are nonempty,

(iii) M ≃ D(M_{∂}, ϵ) ⋃_{E} D(G(p_{1})), where E = ∂D(G(p_{1})) ≃ S(M_{∂}, ϵ),

(iv) M_{∂}/G is homeomorphic to S_{p_{1}}^{⊥}/G_{p_{1}}.

Remark 1.10. The key point in (1.9) is that curv(X) > 0, not sec(M) > 0. If we have only curv(X) ≥ 0, we can apply similar arguments with somewhat weaker conclusions, since dist(∂X, ·) is now only concave, rather than strictly concave. Hereafter, we will refer to the orbit, G(p_{1}), in (1.9) as the “soul”-orbit of G.

Another context in which orbit spaces with non-empty boundary play a significant role is in the following result from [?] (for related result cf. [?]).

Fixed point Lemma 1.11. Let M be a positively curved almosteffective G-manifold with G connected and with principal isotropy subgroup H. If the connected component, H_{0}, of H is a maximal connected subgroup of G, and ∂((G/H_{0})/H_{0}) ≠ ∅, then either M^{G} ≠ ∅, or else G acts transitively on M.

Remark 1.12. The condition ∂((G/H_{0})/H_{0}) ≠ ∅ occurs quite frequently: for example (G, H_{0}) a symmetric pair will satisfy this condition, as will often (G, H_{0}), where H_{0} is maximal. Note however that this is not true for example with (Sp(2), Sp(1)), where Sp(1) is maximal.

In Section 3 we will also need some basic facts about closed manifolds of cohomogeneity one which we recall here for convenience. If dim(M/G) = 1, then M/G is either a circle or an interval. In the first case, all orbits are principal and π : M → X = M/G is a fibration. Since we are interested in positively curved manifolds here, only the second case can arise by the Bonnet-Myers theorem. All interior points of the interval correspond to the principal orbits, E = G/H, and the endpoints of the interval correspond to two exceptional orbits B_{i} = G/K_{i}, i = 0, 1. In terms of this data, M is exhibited as the union of tubular neighborhoods DB_{i} → B_{i}, i = 0, 1, with common boundary ∂DB_{0} ≃ ∂DB_{1} ≃ E. In particular, π_{i} : E = G/H → G/K_{i} = B_{i}, i = 0, 1, are bundles with sphere fibers K_{i}/H = S^{l_{i}}.

Conversely, given

(1.13)  \begin{aligned}& & K_{0} \\ & ⊂ & & ⊂ \\ H & & & & G, K_{i}/H = S^{l_{i}}, i = 0, 1, \\ & ⊂ & & ⊂ \\ & & K_{1} \end{aligned}

we can reconstruct a cohomogeneity one G-manifold as

(1.14)  M = (G ×_{K_{0}} D^{(l_{0}+1)}) ⋃_{G/H}(G ×_{K_{1}} D^{(l_{1}+1)}).

Note that given the isomorphism classes of bundles DB_{i} → B_{i}, different possibilities for M can arise via different glueing maps ∂DB_{0} ≃ ∂DB_{1}. Such glueing maps are G-equivariant, and are determined by an element n ∈ N(H) (cf. (2.6)). In the description (1.13) above, this simply corresponds to replacing only one of the K_{i}’s by its conjugate nK_{i}n^{−1}. We further note that under the assumption that the manifold in question is simply-connected, H is connected, when K_{i}/H ≠ S^{1}, i = 0, 1. The case of finite extensions H of H_{0} and (possibly of) K_{i}, i = 0, 1 in G is possible only if one of the K_{i}/H is a circle. Before we confine our investigation to specific groups, we state one more useful general fact.

Synge (type) Lemma 1.15. Let M be a positively curved manifold and V and W two non-intersecting, totally geodesic submanifolds of M. Then dim(V) + dim(W) < dim(M).

In our context, the submanifolds V and W in (1.15) will arise as fixed point sets for transformation groups K ⊂ G. Although we will not use it here, we remark that (1.15) holds for orbits spaces as well, and even general Alexandrov spaces (cf. [?]).

We point out that the utility of the methods developed in this section is amplified by the obvious fact that they also apply to all subgroups of a given transformation group. Since we are primarily interested in large groups, this will play a significant role as we proceed.

2. Fixed point homogeneous manifolds

In this section we will classify (up to equivariant diffeomorphism) positively curved, fixed point homogeneous manifolds, that is, manifolds, M, for which the fixed point cohomogeneity

(2.1)  cohom fix(M, G) = dim(M/G) − dim(M^{G})

is minimal, i.e., equal to 1.

We need only consider the case in which M^{G} ≠ ∅. If B_{0} is a component of M^{G} with maximal dimension, i.e., dim(B_{0}) = dim(M^{G}), then clearly the codimension of B_{0} in X = M/G is one more than the cohomogeneity of any normal sphere to B_{0} under the induced G-action. Thus, if cohomfix(M, G) = 1, we see that G acts transitively on the normal spheres to B_{0}. In particular, B_{0} is a component of ∂X. Moreover, for ϵ > 0 sufficiently small, the ϵ-neighborhood of B_{0} in X is a smooth manifold with boundary B_{0}, and all orbits in B(B_{0}, ϵ) − B_{0} are principal. As a special case of the Structure Lemma (1.9), we derive the following (cf. also (1.4)):

Structure Theorem 2.2. Let M be a positively manifold with an (almost) effective, isometric G-action of fixed point cohomogeneity one and M^{G} ≠ ∅. If B_{0} is a component of M^{G} with maximal dimension, then the following hold:

(i) There is a unique orbit, B_{1} = G(p_{1}) ≃ G/G_{p_{1}}, at maximal distance to B_{0} (the “soul” orbit).

(ii) All orbits in M − (B_{0} ⋃ B_{1}) are principal and diffeomorphic to S^{k} ≃ G/H, the normal sphere to B_{0}.

(iii) There is a G-equivariant decomposition of M, as

M = DB_{0} ⋃_{E} DB_{1},

where DB_{0}, DB_{1} are the normal disc bundles of B_{0}, B_{1}, respectively, in M, with common boundary E when viewed as tubular neighborhoods.

(iv) All G_{p_{1}}-orbits in the normal sphere S^{l} to B_{1} at p_{1} are principal and diffeomorphic to G_{p_{1}}/H. Moreover, B_{0} is diffeomorphic to S^{l}/G_{p_{1}}.

We leave the details of the proof to the reader and point out only that if dimB_{0} > 0 then B_{0} = ∂X. However, if M^{G} is finite, and hence B_{0} is a point, then X is an interval and B_{0} is one of the boundary points. The other boundary point is either another fixed point for G (in fact, the only other one), or else the orbit at maximal distance from B_{0} (cf. (1.4)). In either case, (2.2) holds as stated.

Note that implicitly in (2.2), we have exhibited two spherical fiber bundles:

K/H → S^{l} → S^{l}/K ≃ B_{0}, (2.3)

K/H → S^{k} ≃ G/H → G/K ≃ B_{1}, (2.4)

where K = G_{p_{1}} is the isotropy group of the soul orbit. This already imposes severe topological restrictions due to Browder [?], from which it is possible to deduce that, at least cohomologically, any M as in (2.2) looks like a finite quotient of a rank-one symmetric space. Utilizing the restrictions on the pair (G, H) expressed in (2.4), we will in fact obtain such a description (Theorem (2.8)) up to (equivariant) diffeomorphism. To acheive this, we need the following:

Uniqueness Lemma 2.5. Let M and M̂ be two (Riemannian) Gmanifolds with structure as in (2.2), i.e., there exist components B_{0} ⊂ M^{G}, B̂_{0} ⊂ M̂^{G} and orbits B_{1} ⊂ M, B̂_{1} ⊂ M̂, such that (ii)-(iv) of (2.2) hold. If, in addition, the bundles DB_{1} → B_{1} and DB̂_{1} → B̂_{1} are G-equivalent, then M and M̂ are G-diffeomorphic.

Proof. We will show that under the assumptions above, any G-equivariant bundle isomorphism f : DB_{1} → DB̂_{1} extends to a Gequivariant diffeomorphism from M to M̂. To do this, we will extend the restriction h = f| : E → Ê to a G-equivariant bundle map g : DB_{0} → DB̂_{0}. Namely, let g be the unique radial extension of h. Then it clearly follows that, F = g ⋃_{h} f : DB_{0} ⋃_{E} DB_{1} → DB̂_{0} ⋃Ê D̂B_{1} is a G-equivariant homeomorphism, F | : M − B_{0} → M̂ − B̂_{0} is a diffeomorphism, and so is F | : B_{0} → B̂_{0} (in fact g|B_{0} ≃ h/G). To check that g : DB_{0} → DB̂_{0} is a diffeomorphism, it therefore suffices to see that it is linear on each fiber. Since isometries of the standard sphere S^{k} = G/H are restrictions of linear maps of R^{k+1} ⊃ D^{k+1}, the desired linearity is an immediate consequence of the following useful fact:

Sublemma 2.6. Let G be a connected, compact Lie group, and H a closed subgroup. Then for any G-equivariant map F : G/H → G/H, there is an n ∈ N(H) ⊂ G such that

F (gH) = gnH.

Moreover, F is an isometry for any homogeneous metric on G/H which is induced from an Ad(N(H))-invariant metric on G.

Proof. Set F (H) = nH. Then F (gH) = gnH for all g ∈ G, by equivariance. However, this is only well-defined if nHn^{−1} = H. The following simple calculation:

dist(F (g_{1}H), F (g_{2}H) = dist(g_{1}nH, g_{2}nH)
= dist(n^{−1}g_{1}^{−1}g_{2}nH, H)
= dist(g_{1}^{−1}g_{2}H, nHn^{−1})
= dist(g_{2}H, g_{1}H)

proves the isometry claim. q.e.d.

In order to fully exploit the Structure Theorem (2.2) and the Uniqueness Lemma (2.5), we shall now use the restrictions imposed on G by the requirement that G/H = S^{k} (cf. also (2.4)). In fact, using the classification of groups that can act transitively on spheres (cf. [?], [?], [?] and [?]), we can assume, (by possibly replacing G by a subgroup) that the pair (G, H) is one of the following:

(2.7) \begin{cases}(a_{k+1})(G, H) = (SO(k + 1), SO(k)) & (k ≥ 1), \\ (b_{m+1})(G, H) = (SU(m + 1), SU(m)) & (k = 2m + 1 ≥ 3), \\ (c_{m+1})(G, H) = (Sp(m + 1), Sp(m)) & (k = 4m + 3 ≥ 7), \\ (d)(G, H) = (G_{2}, SU(3)) & (k = 6), \\ (e)(G, H) = (Spin(7), G_{2}) & (k = 7), \\ (f)(G, H) = (Spin(9), Spin(7)) & (k = 15). \end{cases}

The strategy is now to assume that M is a fixed point homogeneous, positively curved G-manifold, where G is one of the groups in (2.7), and H is the corresponding principal isotropy subgroup. In each case, we determine all potential “soul”-isotropy groups K, such that H ⊂ K ⊂ G satisfies (2.4). For those K which cannot be excluded on the basis of the Structure Theorem (2.2), we find an explicit model M̂ with the same slice representation at the soul orbit and then apply the Uniqueness Lemma (2.5).

We know that if cohomfix(M, G) = 1, M^{G} ≠ ∅ and G is one of the groups listed in (2.7), then codim(M^{G}) = n, 2n, 4n, 7, 8, or 16 corresponding to the cases (a_{n}), (b_{n}), (c_{n}), (d), (e), or (f), respectively. We have used this fact in the formulation of our first main result.

Classification Theorem 2.8. Let M be a closed, connected, fixed point homogeneous Riemannnian manifold. Then M supports an effective and isometric G-action, where G is one of the groups SO(n), SU(n), Sp(n), G_{2}, Spin(7), or Spin(9) and codimM^{G} = n, 2n, 4n, 7, 8, or 16, respectively. If moreover, sec(M) > 0, then M is G-equivariantly diffeomorphic to one of the following:

(a_{n}) S^{m}, RP^{m} (m ≥ n), or in addition, when n = 2, S^{m}/Z_{q} (q ≥ 3) or CP^{m};

(b_{n}) S^{m}, S^{m}/Z_{q} (m ≥ 2n) or CP^{m} (m ≥ n), or in addition, when n = 2, S^{m}/Γ (Γ ⊂ SU(2),(m ≥ 5)), CP^{m}/Z_{2} (m odd) or HP^{m};

(c_{n}) S^{m}, S^{m}/Γ (Γ ⊂ Sp(1), m ≥ 4n), CP^{m} (m ≥ 2n), CP^{m}/Z_{2} (m > 2n odd) or HP^{m} (m ≥ n);

(d) S^{m}, or RP^{m} (m ≥ 7);

(e) S^{m} or RP^{m} (m ≥ 8); or (f) S^{m}, RP^{m} (m ≥ 16) or CaP^{2},

where G in case (a_{n}) is SO(n), etc. as in (2.7).

Proof. First note that the least restrictive cases are (a_{2}), (b_{2}) = (c_{1}) and (f). By abuse of formalism, this is because (a_{k}) ⇒ (a_{k+1}), (b_{l}) ⇒ (b_{l+1}), (c_{m}) ⇒ (c_{m+1}), and (b_{3}) ⇒ (d) ⇒ (e) by standard representation theory. Since (a_{2}) was proven in [?], we will discuss only the cases (b_{2}) and (f) here, and leave the remaining more restrictive cases to the reader.

Case (b_{2}). Let B_{0} be a component of M^{SU(2)} with codimB_{0} = 4, and B_{1} the corresponding soul orbit. SU(2) acts freely on M −(B_{0} ⋃ B_{1}), and the structure of M is determined by the slice representation of the isotropy group K = G_{p_{1}} at p_{1} ∈ B_{1}, according to (2.2) and (2.5). For K, there are the following possibilities:

(i) K = SU(2),

(ii) K = T^{1} = S^{1}, (ii)′ K = N(T^{1}) (N(T^{1})/T^{1} ≃ Z_{2}),

(iii) K = {1}, (iii)′ K = Γ (Γ finite).

Subcase (i). B_{1} = {p_{1}} ⊂ M^{SU(2)}, and SU(2) acts freely on the tangent sphere S^{l} to M at p_{1}. Consequently, l = 4m − 1, the action of SU(2) = Sp(1) on T_{p_{1}}M ≃ R^{4m} ≃ H^{m} is the Hopf action (cf. [?, Sec. 5]), and B_{0} ≃ S^{l}/SU(2) = HP^{m−1}. Now take M̂ = HP^{m} with the obvious SU(2) = Sp(1)-action fixing B̂_{1} = HP^{m−1} and the point p̂_{1} = B̂_{1} at maximal distance from HP^{m−1}. By the Uniqueness Lemma (2.5), M is SU(2)-diffeomorphic to HP^{m}.

Subcase (ii). B_{1} = SU(2)/S^{1} ≃ CP^{1} and K = S^{1} acts freely on the normal sphere S^{l} to B_{1} in M at p_{1}. In particular, l = 2m − 3 and S^{2m−3} → S^{2m−3}/S^{1} = CP^{m−2} ≃ B_{0} is the Hopf map. By the slice theorem, the normal bundle DB_{1} ⊂ V_{1} → B_{1} to B_{1} in M is isomorphic to

SU(2) ×_{S^{1}} R^{2m−2} → SU(2)/S^{1}.

Now take M̂ = CP^{m} with the natural action of U(m + 1). The SU(2)- action on M̂ given via the standard inclusion SU(2) ⊂ U(m + 1) obviously fixes B̂_{0} = CP^{m−2} and acts canonically on B̂_{1} = CP^{1} at maximal distance from CP^{m−2}. By the slice theorem, the normal bundle to B̂_{1} in M̂ is SU(2)-equivalent to the normal bundle of B_{1} in M. Hence by (2.5), M and CP^{m} are SU(2)-diffeomorphic.

Subcase (iii). B_{1} ≃ SU(2), and SU(2) acts freely on M − B_{0}. Since K = {1}, B_{0} ≃ S^{l}, the normal sphere to B_{1} in M at p_{1}. Again by the slice theorem, the normal bundle DB_{1} ⊂ V_{1} → B_{1} of B_{1} in M is trivial. Now take M̂ = S^{l}∗SU(2) = S^{l+4} with the obvious SU(2)-action fixing B̂_{0} = S^{l} and acting by left multiplication on B̂_{1} = SU(2) ≃ S^{3} at maximal distance from S^{l} in S^{l+4}. By (2.5), M is SU(2)-diffeomorphic to S^{l+4}.

Subcase (ii)′. If B_{1} = SU(2)/N(T^{1}) ≃ CP^{1}/Z_{2}, we argue as in (ii) that l = 2m − 3 and B_{0} ≃ S^{2m−3}/N(T^{1}) = CP^{m−2}/Z_{2}. This, however, is only possible if m−2 is odd. In that case, there is indeed an action of SU(2) on M̂ ≃ CP^{m}/Z_{2} which models M in the sense of (2.5). In fact, if τ : CP^{m} → CP^{m} is the involution defining CP^{m}/Z_{2}, then the SU(2)- action on CP^{m} described above takes τ-orbits to τ-orbits (in homogeneous coordinates,

τ([z_{1}, ..., z_{2n}; z_{2n+1}, z_{2n+2}]) = [\overline{z_{n+1}}, ..., \overline{z_{2n}}, −\overline{z_{1}}, ..., −\overline{z_{n}}; \overline{z_{2n+2}}, −\overline{z_{2n+1}}]

if m = 2n + 1).

Subcase (iii)′. If B_{1} = SU(2)/Γ, the finite subgroup

K = Γ ⊂ SU(2)

acts freely on the normal sphere S^{l} to B_{1} in M at p_{1}, and B_{0} ≃ S^{l}/Γ. The normal bundle to B_{1} in M is isomorphic to

SU(2) ×_{Γ} R^{l+1} → SU(2)/Γ.

by the slice theorem. Now consider M̂ = S^{l} ∗ SU(2)/Γ ≃ S^{l+4}/Γ where Γ acts on S^{l} as above and on SU(2) by right translations. The SU(2)- action on S^{l} ∗ SU(2) described in (iii) induces an action on M̂ with B̂_{0} = S^{l}/Γ. Since the normal bundles of B_{1} in M and of B̂_{1} in M̂ are isomorphic, we are done by (2.5).

Case (f). Let B_{0} be a component of M^{Spin(9)} with codim(B_{0}) = 16, and B_{1} = Spin(9)(p_{1}) be the corresponding soul orbit. Since the principal isotropy subgroup is H = Spin(7), and H is necessarily embedded in Spin(8) ⊂ Spin(9) via the spin representation (ref?), there are only the following possibilities for potential soul isotropy subgroups K:

(i) K = Spin(9)

(ii) K = Spin(8);

(ii)′ K = N(Spin(8)) (N(Spin(8)/Spin(8) = Z_{2})

(iii) K = Spin(7);

(iii)′ K = N(Spin(7)) ⊂ Spin(8) (N(Spin(7))/Spin(7) = Z_{2})

Subcase (i). B_{1} = {p_{1}} ⊂ M^{Spin(9)}, and all orbits of the Spin(9)- action on the tangent sphere S^{l} ⊂ T_{p_{1}}M are principal and diffeomorphic to S^{15}. Since there is no proper fibration of a sphere with S^{15} as fiber (cf. e.g. [?]), we conclude that l = 15 and B_{0} ≃ Spin(9)/Spin(7) = S^{15}. Taking M̂ = S^{16}, the suspension of Spin(9)/Spin(7) = S^{15}, we see via (2.5) that M is Spin(9)-equivalent to S^{16}.

Subcase (ii). B_{1} ≃ Spin(9)/Spin(8) = S^{8}, and Spin(8) acts on the normal sphere S^{l} to B_{1} at p_{1}, such that all orbits are principal and diffeomorphic to Spin(8)/Spin(7) = S^{7}. Moreover, B_{0} ≃ S^{l}/Spin(8). Thus either l = 7 and B_{0} = {p_{0}}, or l = 15 (cf. e.g. [?]). However, as shown in [?, p.236], there is no fibration of S^{15} with S^{7} fibers, all of which are also orbits of a group action on S^{15}. Hence l = 7 and the normal sphere bundle E → B_{1} is Spin(9)-equivalent to the Hopf fibration

S^{7} =Spin(8)/Spin(7) → Spin(9)/Spin(7)
=S^{15} → S^{8} = Spin(9)/Spin(8).

The same picture is apparent for the sub-action of Spin(9) ⊂ F_{4} on M̂ = CaP^{2}. Therefore M is Spin(9)-equivalent to CaP^{2} by (2.5).

Subcase (iii). B_{1} = Spin(9)/Spin(7) = S^{15}, and all orbits in M − B_{0} are diffeomorphic to S^{15} and principal. In particular, B_{0} ≃ S^{l}, where S^{l} is the normal sphere to B_{1} in M at p_{1}. Furthermore, the normal bundle to B_{1} in M is Spin(9)-isomorphic to the trivial bundle,

Spin(9)/Spin(7) × R^{l+1} → Spin(9)/Spin(7),

where the action on R^{l+1} is trivial. Pick M̂ = S^{l} ∗ Spin(9)/Spin(7) = S^{l+16} with the obvious Spin(9)-action fixing B̂_{0} = S^{l} and acting canonically on Spin(9)/Spin(7) = S^{15}. Via (2.5), we see that M is Spin(9)- diffeomorphic to S^{l+16}.

Subcase (ii)′. If B_{1} = Spin(9)/N(Spin(8)) ≃ S^{8}/Z_{2}, we see, as in subcase (ii) above, that all N(Spin(8))-orbits in the normal sphere S^{l} to B_{1} at p_{0} must be diffeomorphic to N(Spin(8))/Spin(7) = S^{7} ∐ S^{7}. This excludes l = 7, and we exclude l = 15 as in subcase (ii). Thus K = N(Spin(8)) cannot occur as a soul isotropy subgroup.

Subcase (iii)′. If B_{1} = Spin(9)/N(Spin(7)) = S^{15}/Z_{2} = RP^{15}, an argument, as in subcases (iii) and (ii)′, above shows that M is Spin(9)-diffeomorphic to S^{l+16}/Z_{2} = RP^{l+16}. q.e.d.

Theorem A in the introduction is now an immediate corollary of Theorem 2.8.

3. Low-dimensional non-linear representations

The classification of positively curved manifolds with maximal symmetry rank [?], can also be viewed as a classification of the lowestdimensional manifolds of positive curvature on which a given torus, T^{k}, can act (almost) effectively by isometries.

The principal issue in this section is to analyse the same question for the compact, connected simple Lie groups. Since we allow actions to be almost-effective, it suffices to consider simply-connected groups. Explicitly, the groups we are considering are: Sp(n)(n ≥ 2)), SU(n)(n ≥ 2), Spin(n)(n ≥ 7), together with the exceptional groups G_{2}, F_{4}, E_{6}, E_{7} and E_{8}.

Based on (0.2) in the introduction, we define inductively

(3.1)

rep_{i+1}^{+}(G) = min{n > rep_{i}^{+}(G)|Gacts (almost) effectively

by isometries on some M^{n} with sec(M) > 0}

If we restrict our attention to irreducible linear representations, i.e., M^{n} = S_{1}^{n}, we use the notation rep_{0}^{S}(G) < rep_{1}^{S}(G) < ⋯, and it is obvious that rep_{0}^{+}(G) ≤ rep_{0}^{S}(G) for any compact Lie group G.

Since Sp(n + 1)/Sp(n)Sp(1) = HP^{n}, SU(n + 1)/S(U(n)U(1)) = CP^{n}, Spin(n+1)/Spin(n) = S^{n}, G_{2}/SU(3) = S^{6}, F_{4}/Spin(9) = CaP^{2} all have positive curvature and each one is of the form G/H where dim(H) < dim(G) is maximal, we read off the following simple fact:

Proposition 3.2. If G is one of the simply connected simple groups other than E_{6}, E_{7} or E_{8}, then

rep_{0}^{+}(G) = dimG/H,

where H ⊂ G is a proper subgroup of maximal dimension.

In contrast, it is well known that E_{6}, E_{7} and E_{8} cannot act transitively on a positively curved manifold. Hence from (1.11) we conclude that if G is one of these groups and M is positively curved manifold with dim(M) = rep_{0}^{+}(G), on which G acts (almost) effectively, then the connected component H_{0} of the principal isotropy subgroup H cannot be a maximal connected subgroup. In particular, we conclude

Proposition 3.3. If G is one of E_{6}, E_{7} or E_{8}, then

rep_{0}^{+}(G) ≥ dim(G/H) + 1,

where H is a proper subgroup of G with second lowest codimension.

In particular, this tells us that rep_{0}^{+}(E_{6}) ≥ 33, rep_{0}^{+}(E_{7}) ≥ 56 and rep_{0}^{+}(E_{8}) ≥ 115. Further, we know that the lowest dimensional linear representations of these exceptional groups occur in complex dimensions 27, 56 and real dimension 248 respectively [?]. Thus rep_{0}^{+}(E_{6}) ≤ 52, rep_{0}^{+}(E_{7}) ≤ 110 and rep_{0}^{+}(E_{8}) ≤ 247.

In the remaining part of this section we will classify low-dimensional positively curved manifolds on which the simple groups other than E_{6}, E_{7} or E_{8} can act (almost) effectively by isometries. We first observe that for these groups the lowest-dimensional irreducible linear representations yield transitive actions on the corresponding spheres. Using this fact together with the Fixed Point Lemma (1.11) we obtain:

Fixed Point Corollary 3.4. Let G be a simply-connected, simple Lie group other than E_{6}, E_{7} or E_{8}, and M a positively curved manifold on which G acts (almost) effectively by isometries. If the principal isotropy group H has maximal connected component H_{0} and

dim(M) ≤ min{2rep_{0}^{S}(G) + 1, rep_{1}^{S}(G)},

then cohomfix(M,G)=1.

Another general situation in which fixed point homogeneous manifolds arise naturally occurs because of the simple fact that the principal isotropy group H = G_{p} acts trivially on the normal space to the principal orbit G(p) ≃ G/H at p.

Principal Isotropy Lemma 3.5. Let M be a G-manifold with principal isotropy subgroup H, and isotropy representation

H × T_{H}G/H → T_{H}G/H.

If S_{H} ⊂ T_{H}G/H denotes the unit sphere and cohom fix(S_{H}, K) = 1 for some subgroup K ⊂ H, then cohom fix(M, K) = 1.

The point of this simple fact is that it typically applies to large subgroups H of a simple group G (other than E_{6}, E_{7} or E_{8}).

When applying either (3.4) or (3.5), the cohomogeneity of G×M → M is irrelevant. If, however, the principal isotropy subgroup H ⊂ G is such that neither (3.4), nor (3.5), can be utilized, then we resort to other restrictions imposed by the assumption that

(3.6)  dim(M) = dim(M/G) + dim(G/H)

is relatively small. In particular, we consider only actions where the principal isotropy subgroup H has fairly small codimension in G. This in turn restricts all the possible isotropy subgroups and enhances the chances for using (1.3). So far, however, we have only been able to make systematic use of this approach when, in addition, cohom(M, G) = dim(M/G) ≤ 1.

Theorem 3.7 (Symplectic Groups). Let M be a simply-connected, closed manifold with sec(M) > 0. If Sp(n + 1), n ≥ 1, acts (almost) effectively by isometries on M and

dim(M) ≤ C(n) = \begin{cases}8n − 3 = 2rep_{0}^{+}(Sp(n + 1)), & n ≥ 2 \\ 8 & n = 1, \end{cases}

then dim(M) ≥ 4n = rep_{0}^{+}(Sp(n + 1), and M is diffeomorphic to one of either a sphere, a complex or quaternionic projective space, the flagmanifold Sp(3)/(Sp(1))^{3} or the real homology sphere Sp(2)/SU(2).

Proof. Let H denote the principal isotropy subgroup of

G = Sp(n + 1)

acting on M. From (3.6) and dim(M) ≤ C(n) we have

(3.8)  dim(H) ≥ dim(Sp(n + 1)) − C(n).

An analysis of the possible connected subgroups H_{0} ⊂ Sp(n + 1) satisfying (3.8) yields the following list:

(a)  H_{0} = Sp(n)Sp(1)  n ≥ 1,
(b)  H_{0} = Sp(n)U(1)  n ≥ 1,
(c)  H_{0} = Sp(n)  n ≥ 1,
(d)  H_{0} = Sp(n − 1)Sp(2)  n ≥ 2,
(e)  H_{0} = Sp(n − 1)(Sp(1))^{2}  n ≥ 2,
(f)  H_{0} = U(n + 1)  n ≤ 3,
(g)  H_{0} = SU(n + 1)  n ≤ 3.

In the case (g), we note that for n ≥ 2 the action must be transitive, and since neither space obtained is of positive curvature, these cases do not occur. Note that for n = 1, case (g) coincides with case (c) (as Sp(1) = SU(2)) and we will treat it later. In the cases (d) and (f), we can use Corollary (3.4), and hence (2.8) since (G, H_{0}) is a symmetric pair and rep_{0}^{S}(Sp(n+1)) = 4n+3, rep_{1}^{S}(Sp(n+1) = (n+1)(2(n+1)−1)−1, n ≥ 2 by standard representation theory. For case (f) (n = 1), we note that the two lowest-dimensional irreducible linear representations of Sp(2) occur in dimensions 5 and 8, but both are transitive on the corresponding spheres. Thus, cohomfix(M, Sp(2)) = 1 with H_{0} = U(2) ⊂ Sp(2).

Moreover, except for Sp(1) in case (c) (n = 1), all of these groups admit only one embedding in Sp(n + 1), up to conjugation. Aside from the standard embedding Sp(1) = Sp(1) × {1} ⊂ Sp(1) × Sp(1) ⊂ Sp(2), we can embed Sp(1) via the diagonal Sp(1) = Δ(Sp(1)×Sp(1)) ⊂ Sp(2) and as a maximal subgroup Sp(1) ⊂ Sp(2). The diagonal embedding can also be viewed as Sp(1) = SU(2) ⊂ U(2) ⊂ Sp(2).

In the first three cases, in which the embedding is standard, i.e., (a) and (b) for all n, and (c) for n ≥ 2, we apply the Principal Isotropy Lemma (3.5) to the subgroup K = Sp(n) ⊂ H_{0} ⊂ H, and then appeal to the Classification Theorem (2.8).

In order to complete case (c), it remains to consider the case where the embedding of H_{0} is not standard. The only possible dimensions for M are then 7 or 8. If dim(M) = 7, the Sp(2)-action is transitive and M = Sp(2)/Sp(1) is the Berger homology sphere [?]. If dim(M) = 8, the Sp(2)-action is of cohomogeneity one. In particular, the only case consistent with (1.13) is that in which the embedding of H_{0} is diagonal, and we will rule out this case. Note that the only possible subgroups K_{0}, K_{1} between Sp(1) ≃ SU(2) and Sp(2) satisfying the conditions of (1.13) are Sp(1)^{2} and U(2).

Assume first that K_{0} = K_{1} = U(2). We remark further that we may exclude (by means of a general argument) the case in which K_{0} = K_{1} ≠ G [?] for a cohomogeneity one manifold of positive curvature. However, for the sake of comleteness we will prove each individual case as it arises (cf. (3.7) case (e), (3.9) cases (d) and (h), (3.11) case (c) and (3.13)). Then the principal orbit E = Sp(2)/Sp(1) fibers over the exceptional orbits B_{0} ≃ B_{1} ≃ Sp(2)/U(2) with common fibers S^{1} ≃ U(2)/Sp(1), and M fibers over B_{0} ≃ Sp(2)/U(2) with fiber S^{2}. Moreover, E^{SU(2)} ⊂ E consists of two disjoint circles (namely the orbit of N(SU(2))), each of which is a fiber over B_{0} and B_{1}. The corresponding S^{2}-fibers over B_{i} in M are also fixed by H = SU(2), and in fact they are components of M^{SU(2)}. The latter fact is seen via the isotropy representation of U(2)(⊃ SU(2)) at the corresponding fixed points in B_{i}, i = 0, 1. Now fix S^{1} ⊂ SU(2) ⊂ U(2), and consider M^{S^{1}}. At each of the fixed points p_{i}, q_{i} ∈ B_{i}, i = 0, 1, for U(2) acting on B_{i}, the isotropy representation of U(2) reveals that the corresponding components of M^{S^{1}} are 4-dimensional. By the Synge Lemma (1.15), they must all be contained in the same component, which, however, clearly contains the above (disjoint) S^{2}- components of M^{SU(2)}, impossible again by (1.15).

From the above, we conclude that one of the K_{i}’s is Sp(1) × Sp(1), and it is not difficult to see that the action of Sp(2) ≃ Spin(5) is not effective on M as exhibited in (1.14). The corresponding effective action is by G = SO(5) with principal isotropy subgroup H = SO(3) embedded in the standard way. Moreover, K_{0}, K_{1} are either SO(4) or SO(3)×SO(2). If K_{0} ≃ K_{1} = SO(4) we proceed as follows. H = SO(3) fixes two disjoint circles in E = G/H, each of which is mapped to one circle in B_{0} and in B_{1}. Indeed, M^{SO(3)} is a torus. This is a contradiction, since it is also totally geodesic and hence positively curved.

To complete case (c), it remains to consider the cohomogeneity one action on M^{8} by G = SO(5), with H = SO(3), K_{0} = SO(3)SO(2) and K_{1} = SO(4). First observe that the SO(3)-factor of K_{0} fixes a totally geodesic S^{2} in M (that is, two points in B_{0}, two circles in E and one circle in B_{1}). Fix L = SO(2) ⊂ SO(3) and consider M^{L}. From the isotropy representations, we see that M^{L} is a 4-manifold. Its intersection with B_{1} is a 2-sphere. Now the SO(2)-factor of K_{0} acts on M^{L} preserving M^{SO(3)}. It has exactly four fixed points (2 in B_{0} and 2 in B_{1}), which is impossible if M and hence M^{L} have positive curvature, by the Extent Lemma (1.3) (cf. [?]).

Note that we must also worry about finite extensions of H_{0} and K_{i}, i = 0, 1, in this case, since K_{i}/H can be a circle for at least one i. In the case where K_{0} = K_{1} = U(2), we may extend H_{0} to H_{0} × Z_{k} = H. However, the argument used to exclude the case with H connected works as well in this case. In the case where K_{0} = SO(3)SO(2) and K_{1} = SO(4), any finite extension of H_{0} = SO(3) in G must include the corresponding finite extension of SO(4), since otherwise K_{1}/H will not be a sphere. This leaves us with only one possibility: H = SO(3)×Z_{2} = O(3) and K_{1} = O(4). In this case, M = CP^{4} (cf. [?].

We now turn to the remaining case (e). We will show that only n = 2 can occur, and in that case M is either homogeneous, i.e., M = Sp(3)/Sp(1)^{3}, or it has cohomogeneity one and M = S^{13}. In fact, for all n ≥ 2 in the given range of dimensions for M, M must either be homogeneous or of cohomogeneity one. The classification of homogeneous manifolds with positive curvature leaves only the flagmanifold in dimension 12 above as a possibility. On the other hand, if M has cohomogeneity one, its data is given according to (1.13) as:

K_{0} = Sp(n − 1)Sp(2)
⊂  ⊂
H = Sp(n − 1)Sp(1)Sp(1)  G = Sp(n + 1)
⊂  ⊂
K_{1} = Sp(n − 1)Sp(2)

where K_{i}/H ≃ Sp(n−1)Sp(2)/Sp(n−1)(Sp(1))^{2} are 4-spheres for both i. Moreover, for n ≥ 3 there is only one possible embedding of K_{0} = K_{1}. For n = 2, however, we can also embed K_{0} ≃ K_{1} by permuting the factors. This is exactly the description of S^{13} under the representation ∧^{2}ν − θ of Sp(3) (notation from [?]). It remains to show that K_{0} = K_{1} cannot occur when sec(M) > 0.

The general case n ≥ 3 reduces to the case n = 2, since it is easy to see from the isotropy representations that M^{8n−3} contains a totally geodesic submanifold N^{13} of dimension 13, which is fixed by Sp(n−2) ⊂ Sp(n−2)(Sp(1))^{3}, and on which Sp(3) acts by cohomogeneity one, with H = (Sp(1))^{3} and K_{0} = K_{1} = Sp(1)Sp(2). Thus, it suffices to show that the case n = 2 in which K_{0} = K_{1} cannot occur.

Here, we see that the principal orbit E = Sp(3)/(Sp(1))^{3} fibers over the two exceptional orbits B_{0} ≃ B_{1} ≃ Sp(3)/Sp(1)Sp(2) = HP^{2} with common fibers S^{4}. In particular, M^{13} fibers over HP^{2} with fiber S^{5}. Let p_{i} ∈ B_{i}, i = 0, 1, be the fixed points of K = K_{0} = K_{1} = Sp(1)Sp(2) on B_{i}, and S^{4} ≃ HP^{1} ≃ N_{i} ⊂ B_{i} the K-orbits in B_{i} at maximal distance from p_{i} in B_{i}. Note that the Sp(1)-factor of K acts trivially on N_{i}, and that the Sp(2)-factor acts transitively on N_{i} with principal isotropy Sp(1)Sp(1). For the Sp(1)-factor of K, consider M^{Sp(1)}. At p_{i} ∈ B_{i}, Sp(1) acts freely on the tangent sphere of B_{i} and trivially on the normal sphere. In particular, the 5-sphere fiber of M → B_{i} suspended between p_{0} and p_{1} is totally geodesic, and a component of M^{Sp(1)}. Now consider the action of Sp(1) at points in N_{i} ⊂ B_{i}. On the normal sphere to N_{i} inside B_{i}, the action is free. Thus the component of M^{Sp(1)} containing N_{i} is determined by the action of Sp(1) normal to B_{i} at N_{i}. From representation theory, this Sp(1)×S^{4} → S^{4} action is either almost-effective and factors through SO(3), or it is the suspension of the standard free action on S^{3}. In the first scenario, we find disjoint totally geodesic submanifolds of M of dimensions 9 and 5, contradicting (1.15). In the second scenario, we find a 5-dimensional component, V^{5}, of M^{Sp(1)} containing the B_{i}’s. Moreover, the Sp(2)-factor of K acts on V^{5} with cohomogeneity one, and all orbits are of principal type S^{4} ≃ B_{i}. In particular, V^{5} fibers over S^{1}, which is impossible, since V^{5} has positive curvature. q.e.d.

Theorem 3.9 (Unitary Groups). Let M be a simply-connected, closed manifold with sec(M) > 0. If SU(n + 1), n ≥ 1 acts (almost) effectively by isometries on M and

dim(M) ≤ C(n) = \begin{cases}4n − 2 = 2rep_{0}^{+}(SU(n + 1)), & n ≥ 3, \\ 7 & n = 2, \\ 4 & n = 1, \end{cases}

then dim(M) ≥ 2n = rep_{0}^{+}(SU(n + 1)), and M is diffeomorphic to one of the following: a sphere, a complex projective space, the flagmanifold SU(3)/T^{2}, an Aloff-Wallach space SU(3)/S_{k,l}^{1} or the Berger manifold SU(5)/Sp(2)S^{1}.

Proof. As in the proof of (3.7), we list all the possibilities for the connected component, H_{0}, of the principal isotropy subgroup, H, under the restriction dim(M) ≤ C(n). The list is:

(a)  U(n) = S(U(n)U(1))  (n ≥ 1),
(b)  SU(n)  (n ≥ 2),
(c)  S(U(n − 1)U(2))  (n ≥ 3),
(d)  SU(n − 1)SU(2)  (n ≥ 3),
(e)  SO(n + 1)  (2 ≤ n ≤ 4),
(f)  Sp (\frac{n + 1}{2})  (n = 3, 5),
(g)  Sp(2)S^{1}  (n = 4),
(h)  T^{2}  (n = 2),
(i)  S^{1}  (n = 1, 2),
(j)  {1}  (n = 1).

In all cases, with the exception of case (i) (n = 2), there is only one embedding of H_{0} in SU(n + 1), up to conjugation. In the first two cases we apply the Classification Theorem (2.8) via (3.5). In fact, in case (a) (n = 1), (M, U(1)) is fixed point homogeneous and so is (M, K), with K = SU(n) in the remaining cases.

The cases (c), (e) and (f) are all done via (3.4). We remark first that in all these cases (G, H_{0}) is a symmetric pair, and secondly that

rep_{0}^{S}(SU(n + 1) = 2(n + 1) − 1

and

rep_{1}^{S}(SU(n + 1)) = n(n + 1) − 1

for n ≥ 4, and for n = 3,

rep_{0}^{S}(SU(4)) = rep_{0}^{S}(Spin(6)) = 5

and rep_{1}^{S}(SU(4)) = 7 (and both of these representations are transitive), and in case (e) (n = 2), we have rep_{0}^{S}(SU(3)) = 5 and rep_{1}^{S}(SU(3)) = 7.

We now proceed to show that case (d) cannot occur. We remark first, that for dimension reasons, the action of SU(n + 1) must either be transitive or of cohomogeneity one. The first option is ruled out by the classification of positively curved homogeneous manifolds, and thus we assume that cohom(M, SU(n + 1)) = 1.

Note first that for n ≥ 4, H_{0} = H and the only possible groups satisfying (1.13) are:

\begin{matrix}H = & SU(n − 1)SU(2) ⊂ S(U(n − 1)U(2)) \\ = & K_{0} = K_{1} ⊂ SU(n + 1) = G, \end{matrix} (3.10)

and N(H) = K = K_{i}, i = 0, 1, K/H = S^{1}. Remark also that, as in the discussion of case (e) in (3.7), the subcases (n ≥ 5) reduce to the subcase (n = 3). We will first rule out the subcase (n = 4). Here, M fibers over B_{0} ≃ B_{1} ≃ G/K ≃ G_{3,2} with fiber S^{2}, and K fixes isolated points p_{i} ∈ B_{i}, i = 0, 1. Let A_{i} ⊂ B_{i} be the K-orbit at maximal distance from p_{i}, i = 0, 1. Then A_{i} ≃ G_{2,1} ≃ CP^{2} is fixed by the SU(2)-factor of H. This implies that SU(2) also fixes the normal bundles of B_{i} in M restricted to A_{i}, i = 0, 1. The resulting G-manifold is a component of M^{SU(2)} and hence totally geodesic. It fibers over A_{i} ≃ CP^{2} with S^{2}-fiber, and the SU(3)-factor of H acts on it by cohomogeneity 1 or 2, either of which gives us a contradiction; the first via the Principal Isotropy Lemma (3.5), and the second via the Fixed Point Lemma (1.11).

To complete case (d), we now proceed with the subcase (n = 3). According to (1.13), the only possibilities for K_{i}, i = 0, 1 are S(U(2)U(2)) ≃ Spin(4)Spin(2) and Spin(5). And the argument in this case mirrors the argument made for case (e) (n = 3) in (3.7). The only possible in this case is M = CP^{5} where G = SO(6)/Z_{2} (cf. [?]). The details are left to the reader.

In case (g), the SU(5)-action is either transitive or of cohomogeneity one. However, there are no subgroups K_{i}, between H_{0} = Sp(2)S^{1} and SU(5), satisfying (1.13). Thus, M^{13} = SU(5)/Sp(2)S^{1}, the Berger example, is the only possibility here.

In case (h) as well, the SU(3)-action is either transitive or of cohomogeneity one. In the homogeneous case, we obtain the flagmanifold, M^{6} = SU(3)/T^{2}. When the action is of cohomogeneity one, we note first that only K_{i} ≃ U(2) satisfies (1.13). Moreover, there are only two choices for the pair (K_{0}, K_{1}): either K_{0} = K_{1} or K_{0} ≠ g^{−1}K_{0}g = K_{1} is embedded via a permutation of the coordinates. The latter case characterizes S^{7}, where SU(3) acts on R^{8} via the adjoint representation. We will show that the former does not occur. First note that M^{7} fibers over B_{0} ≃ B_{1} ≃ SU(3)/K = SU(3)/S(U(2)U(1)) = CP^{2} with S^{3}- fibers, and K fixes isolated points p_{i} ∈ B_{i}, i = 0, 1. Let A_{i} ⊂ B_{i} be the K-orbits at maximal distance to p_{i}. Then A_{i} ≃ U(2)/T^{2} ≃ CP^{2}, and there is an S^{1} ⊂ T^{2} which fixes all of A_{i}. This S^{1} also acts on the normal bundle of B_{i} restricted to A_{i}, and therefore either fixes the whole normal bundle or a 1-dimensional sub-bundle. The latter is impossible, since it would yield a totally geodesic component of M^{S^{1}} of the form S^{2} × S^{1}. If, on the other hand, the whole normal bundle is fixed, we get a 5-dimensional component of M^{S^{1}}, namely the restriction of the S^{3}-fibration M → B_{0} to A_{0}. It is, however, also easy to see that all S^{3}-fibers are totally geodesic, so a contradiction in this case is reached via the Synge Lemma (1.15).

In case (i) (n = 2), the action of SU(3) must necessarily be transitive, and hence M is an Aloff-Walach example SU(3)/S_{k,l}^{1}.

It remains to consider cases (i) and (j), (n = 1), where we have SU(2)-actions on manifolds with dim(M) ≤ 4. If H_{0} = S^{1}, we are done by (3.5). If H_{0} = {1} and dim(M) = 4, the SU(2)-action is of cohomogeneity one. The only possible groups, K_{i}, between {1} and SU(2) satisfying (1.13) are Z_{2}, S^{1} or SU(2). If Z_{2} arises, then π_{1} ≠ {1}, and if SU(2) does, then we are done by the Classification Theorem (2.8). In the case where K_{0} ≃ K_{1} ≃ S^{1}, χ(M) = 4, which is impossible by [?].

Note that we must also worry about finite extensions here, since the principal orbit may fiber over the singular orbit with circle fiber. There are only two such SU(2) actions, both of which are ineffective. The corresponding (ineffective) SO(3) actions have principal isotropy subgroup Z_{2} or Z_{2}×Z_{2}. In the first case M = CP^{2} and the SO(3) action is the restriction of the standard SU(3) action on CP^{2}. In the second case M = S^{4} and the action of SO(3) on S^{4} is via the representation S^{2}ρ_{2} − θ (notation from [?]) [?].

Theorem 3.11 (Orthogonal Groups). Let M be a simply-connected, closed Riemannian manifold with sec(M) > 0. If Spin(n + 1), n ≥ 6 acts isometrically and (almost) effectively on M and

dim(M) ≤ C(n) = 2n = 2rep_{0}^{+}(Spin(n + 1)),

then dim(M) ≥ n = rep_{0}^{+}(Spin(n + 1)), and M is diffeomorphic to a sphere, a complex projective space, or the Cayley plane.

Proof. As in the previous two theorems, we proceed to list the connected components of the possible principal isotropy subgroups, under the given dimensional restrictions. The possibilities are:

(a)  H_{0} = Spin(n),  n ≥ 6,
(b)  H_{0} = Spin(n − 1)S^{1},  n ≥ 6,
(c)  H_{0} = Spin(n − 1),  n ≥ 6,
(d)  H_{0} = SU(4) ≃ Spin(6),  n = 7,
(e)  H_{0} = G_{2},  n = 6.

In all cases except (c) and (d), H_{0} is a maximal connected subgroup of G = Spin(n + 1), and ∂((G/H_{0})/H_{0}) ≠ ∅, since (a) and (b) are symmetric pairs, and in case (e), (G/H_{0})/H_{0} is a closed interval. However, the only positively curved manifolds on which Spin(n + 1) can act transitively are the spheres S^{n} = SO(n + 1)/SO(n), S^{7} = Spin(7)/G_{2} and S^{15} = Spin(9)/Spin(7). Suppose then that Spin(n + 1) does not act transitively on M. Then by the Fixed Point Lemma (1.11), M^{Spin(n+1)} ≠ ∅. The action of Spin(n + 1) at the normal space to a point in M^{Spin(n+1)} yields a representation of dimension less than or equal to 2n by the assumption on dim(M). For n ≠ 6, 8, 9, or 11 the two lowest linear representations of Spin(n + 1) are in dimensions n + 1 and \frac{n(n+1)}{2}, and Corollary 3.4 together with (2.8) yields the desired result. The three lowest dimensional representations for Spin(7) are of dimensions 7, 8, 21; for Spin(9): 9, 16, 36; for Spin(10): 10, 16, 45; and for Spin(12): 12, 64, 66. Since the two lowest-dimensional representations for Spin(7), as well as for Spin(9), yield transitive actions on the corresponding spheres, the Fixed Point Lemma (1.11), together with (2.8), still suffices without further considerations. In the case of Spin(10), the 16-dimensional representation (which is not transitive on S^{15}) might occur when 16 ≤ dim(M) ≤ 18. However, since the principal isotropy subgroup for this action on S^{15} is neither Spin(9), nor Spin(8)S^{1} (known from representation theory, or can be seen via the Fixed Point Lemma applied to Spin(10) × S^{15} → S^{15}), this case does not arise.

We are left then with case (c) and (d). Note however that there is an outer automorphism of Spin(8) (triality) which take SU(4) to Spin(6) and so case (d) is contained in case (c). For n ≠ 8, there is only one embedding of Spin(n − 1) in Spin(n + 1), and Spin(n + 1)/Spin(n − 1) does not carry a homogeneous metric of positive curvature, so for these n, Spin(n+1)×M → M must be of cohomogeneity one and dim(M) = 2n. For the non-standard embedding of Spin(7) in Spin(9), however, Spin(9)/Spin(7) = S^{15}.

Now suppose that Spin(n + 1) acts (almost) effectively on M^{2n} with cohomogeneity one and principal isotropy subgroup H = H_{0} = Spin(n − 1) embedded in the standard fashion (also when n = 8). The possible subgroups K_{0}, K_{1} satisfying (1.13) are then Spin(n − 1)S^{1} and Spin(n). Note also that there is only one embedding of Spin(n − 1)S^{1} in Spin(n+1), whereas Spin(n) admits embeddings parametrized by S^{1} (these, however, all yield the same manifold up to diffeomorphism). As we have seen in previous theorems, it suffices to consider only the subcases (n = 4) and (n = 5), and since Spin(5) = Sp(2) and Spin(6) = SU(4), both have already been ruled out. Thus, case (c) does not occur when H = H_{0} = Spin(n−1) ⊂ Spin(n+1) is standard. If H/H_{0} ≠ {1}, then M = CP^{n} (cf. [?]).

Finally, we consider case (c) (n = 8), where the embedding of Spin(7) in Spin(9) is not standard, i.e., suppose Spin(9) acts on M^{16} by cohomogeneity one, with principal isotropy subgroup H_{0} = Spin(7) embedded via the spin-representation in Spin(8) ⊂ Spin(9). According to (1.13) there are only two possibilities for K_{i}, i = 0, 1, corresponding to 3 possible scenarios: (i) K_{0} = K_{1} = Spin(9), (ii) K_{0} = Spin(9) and K_{1} = Spin(8) and (iii) K_{0} ≃ K_{1} = Spin(8). The first two cases correspond to S^{16} and CaP^{2} respectively, and we show how to rule out the third case here. To do so, we consider the orbit space M/Spin(8). Here G/K_{i} = S^{8}, and the action of Spin(8) on the singular orbits fixes 2 isolated points and is transitive on the normal S^{7} to both of these points, that is, M/Spin(8), (G/K_{i})/Spin(8) =I, i = 0, 1. M^{Spin(8)} consists of four isolated points. Moreover the induced representation of Spin(8) on the tangent spaces R^{16} to the fixed points in M yields a cohomogeneity one action on S^{15} with principal isotropy group G_{2} and orbit space [0, π/2]. A contradiction is thus obtained via the Extent Lemma (1.3).

q.e.d.

The exceptional groups G_{2} and F_{4} have SU(3) and Spin(9) as subgroups of maximal dimension, respectively. Since G_{2}/SU(3) = S^{6} and F_{4}/Spin(9) = CaP^{2} both have (homogeneous) positive curvature metrics, it follows that rep_{0}^{+}(G_{2}) = 6 and rep_{0}^{+}(F_{4}) = 16. Further, it is known that rep_{1}^{S}(G_{2}) = 14 and rep_{0}^{S}(F_{4}) = 25 and rep_{1}^{S}(F_{4}) = 51. By arguments as in the previous three theorems, we derive:

Theorem 3.12. Let M be a positively curved manifold with π_{1}(M) = {1}. If G_{2} acts (almost) effectively on M by isometries and

dim(M) ≤ 11 = 2rep_{0}^{+}(G_{2}) − 1,

then dim(M) ≥ 6 = rep_{0}^{+}(G_{2}), and M is diffeomorphic to a sphere.

Theorem 3.13. Let M be a simply-connected manifold with sec(M) > 0, as above. If F_{4} acts isometrically and (almost) effectively on M and

dim(M) ≤ 25 = 2rep_{0}^{+}(F_{4}) − 7,

then dim(M) ≥ 16 = rep_{0}^{+}(F_{4}), and M is diffeomorphic to a sphere, the Cayley plane, or the flagmanifold F_{4}/Spin(8).

The Corollaries C and D in the introduction now follow easily from Theorems (3.7), (3.9), (3.11), (3.12), and (3.13), and the classification of positively curved homogeneous manifolds.

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University of Maryland Instituto de Matemáticas, Unidad Cuernavaca–UNAM, CINVESTAV-IPN, MEXICO