Positively curved 6-manifolds with simple symmetry groups

Positively curved 6-manifolds with simple symmetry groups

FUQUAN FANG

Nankai Institute of Mathematics, Nankai University, Tianjin 300071, P.R.C.

Instituto de Matemática, Universidade Federal Fluminense, Niterói, RJ, Brasil

Manuscript received on July 16, 2002; accepted for publication on September 3, 2002;

presented by MANFREDO DO CARMO

ABSTRACT

Let M be a simply connected compact 6-manifold of positive sectional curvature. If the identity component of the isometry group contains a simple Lie subgroup, we prove that M is diffeomorphic to one of the five manifolds listed in Theorem A.

Key words: 6-manifolds, positive curvature, symmetry groups.

0 . INTRODUCTION

An interesting problem in Riemannian geometry is to classify positively curved manifolds whose isometry groups are large. A typical example is the classification of homogeneous manifolds of positive curvature (cf. Aloff and Wallach 1975, Berger 1961, Bergery 1976, Grove and Searle 1997).

Recently, there has been considerable progress on classification of positively curved manifolds with large abelian symmetry group actions (cf. Hsiang and Kleiner 1989, Fang and Rong 2001, 2002, Grove and Searle 1994, Rong 2002, Wilking 2002). The maximal rank theorem of Grove-Searle asserts that if a compact positively curved manifold M admits an isometric T^k-action, then and "='' if and only if M is diffeomorphic to a sphere, a lens space or a complex projective space (Grove and Searle 1994). The homeomorphism classification for positively curved n-manifolds with almost maximal rank have been carried out for n = 4 by Hsiang and Kleiner (1989), for n = 5 by Rong (2002) and for n³ 8 completely solved by Fang and Rong (2001). Very recently Wilking (2002) announced the following 1/2-maximal rank theorem: For n³ 10, if a closed simply connected positively curved n-manifold M with symmetry rank (i.e. the rank of its isometry group) at least , then M is homeomorphic to a sphere or a quaternionic projective space or homotopic to a complex projective space (Wilking 2002).

In the geometry of positive sectional curvature, 6-dimensional manifolds play very interesting roles. This is because, there are so called "flag'' manifolds, the homogeneous space SU(2)/T² and the biquotient space SU(3)//T² on which there exist infinitely many circle bundles whose total spaces (dimension 7) admit positive sectional curvature, the Aloff-Wallach spaces and the Eschenberg spaces (cf. Aloff and Wallach 1975, Eschenberg 1982, 1992). Except the two flag manifolds, so far only S⁶ and are known to admit positive curvature metrics. All of the four examples admit a symmetry of a simple Lie group.

The purpose of this paper is to study the isometric G-actions on positively curved manifolds of dimension 6 where G is a simple Lie group. We will prove almost a converse holds, except that one more topology type, S²´ S⁴, maybe occur.

In the below = will indicate manifolds of the same diffeomorphism type.

THEOREM A. Let M be a simply connected compact 6-manifold of positive sectional curvature. If the identity component of the isometry group of M contains a simple Lie subgroup. Then M = S⁶, S² ´ S⁴, , SU(3)/T² or SU(3)//T².

Note that any simple Lie group contains either SU(2) or SO(3) as a subgroup. Therefore one really needs only to study manifolds with symmetry group G = SU(2) or G = SO(3). Thus Theorem A follows immediately from the following two results.

THEOREM B. Let M be a simply connected compact 6-manifold of positive sectional curvature. If M admits an effective isometric SU(2)-action, then M = S⁶, S²´ S⁴,, SU(3)/T² or SU(3)//T².

THEOREM C. Let M be a simply connected compact 6-manifold of positive sectional curvature. If M admits an isometric SO(3)-action (not necessarily effective) then M = S⁶, S²´ S⁴, .

The rest of the paper is organized as follows: In Section 1, we collect some necessary preliminaries in the proof of Theorems B and C. In Section 2, we prove Theorems B and C.

1. PRELIMINARIES

A. POSITIVE CURVATURE

For positively curved simply connected manifolds, few general results are known. In our proof of Theorem A, the following results in this subsection are required.

THEOREM 1.1 (SYNGE 1936). Let M be an n-manifold of sec_M > 0.

(1.1.1) If n is even and M is orientable, then any orientation preserving isometry f has a non-empty fixed point set.

(1.1.2) If n is odd, then M is orientable and any orientation reversing isometry f has a non-empty fixed point set.

THEOREM 1.2 (FRANKEL 1961). Let M be a compact manifold of positive sectional curvature. If two totally geodesic submanifolds N₁ and N₂ satisfies

dim(N₁) + dim(N₂) ³ dim(M),

then N₁ and N₂ have a non-empty intersection.

The asymptotic index of an immersion f : N ® M is defined by v _f = min_{x ÎN} v _f (x), where v _f (x) is the maximal dimension of a subspace of T_xN on which the second fundamental form vanishes. Clearly, f is a totally geodesic immersion if and only if v _f = dim(N).

THEOREM 1.3 (FANG ET AL. 2002). Let M^m be a closed manifold of positive sectional curvature, and let N_j

M^m be a closed embedded submanifold of asymptotic index v_j (j = 1, 2). If either N_j (j = 1, 2) is minimal or N₁ = N₂, then the following natural homeomorphisms,

p_i (N₁, N₁Ç N₂) ® p_i (M^m, N₂), p_i (N₂, N₁Ç N₂) ® p_i (M^m, N₁),

are isomorphism for i £ v₁ + v₂ - m and are surjections for i = v₁ + v₂ - m + 1.

Applying to totally geodesic submanifold, where the asymptotic index is exactly the dimension, Theorem 1.3 implies immediately that

THEOREM 1.4 (WILKING 2002). If Mⁿ has positive sectional curvature and if N^{n - k} is an embedded totally geodesic submanifold then the inclusion map i : N^{n - k}® Mⁿ is n-2k+1 connected, that is i induces an isomorphism of the homotopy groups up to dimension n - 2k and it maps p_n-2k+1 (N) surjectively onto p_n-2k+1 (M).

B. ALEXANDROV SPACES WITH POSITIVE CURVATURE

Recall that an Alexandrov space, X, is a finite Hausdorff dimensional complete inner metric space with a lower curvature bound in distance comparison sense (cf. Burago et al. 1992). In particular, a Riemannian manifold of sectional curvature bounded from below is an Alexandrov space. Typical examples of non-manifold type Alexandrov spaces are given by the following

LEMMA 1.5 (BURAGO ET AL. 1992). Let X be an Alexandrov space with curvature ³ - L . Let G be a compact group of isometries. Then, the quotient space, X/G, is also an Alexandrov space with curvature ³ - L .

2. PROOFS OF THEOREMS B AND C

In this section we consider simply connected positively curved 6-manifolds with G-symmetry where G = SO(3) or SU(2). Note that the connected symmetry group of SU(3)/T² is PSU(3) = SU(3)/ , where is the center of SU(3) (cf. Shankar 2001a). It was pointed out to the author by Shankar (2001b) that the identity component of the symmetry group of the biquotient SU(3)//T² is U(2). In both cases, SU(2) is a proper subgroup of the isometry groups.

LEMMA 2.1. Let M be a simply connected compact 6-manifold of positive sectional curvature. If M admits an isometric SU(2)-action such that the fixed point set Fix(M, SU(2)) is not empty, then M is either diffeomorphic to S⁶ or .

PROOF. Let X = Fix(M, SU(2)). Since an irreducible representation of SU(2) has dimension 4, the fixed point set must be of dimension 2. Therefore the orbifold M/SU(2) is an Alexandrov space with boundary X. Thus

M = E(h ) È (SU(2) ´ _HD^m)

where E(h) is a 4-disk bundle over X, and H Ì SU(2) is the isotropy group for some single orbit, m + 3 = dimH + 6 (cf. Grove and Searle 1997). In particular, X is connected, and so X = S² (since X is totally geodesic with positive curvature). Note that the boundary of E(h ) is S²´ S³.

(i) If H is of dimension 0;

Then the neighborhood SU(2) ´_HD³ must be SU(2) ´ D³. This implies that M is diffeomorphic to S⁶.

(ii) If H is of dimension 1;

Then the neighborhood SU(2) ´_HD⁴ must be SU(2) ´ _S¹D⁴, which is equivariant diffeomorphic to S²´ D⁴ with the product action of SU(2), i.e. on the second factor is given by the irreducible representation of SU(2) on D⁴Ì , on the first factor the action is given by the natural action of SU(2) on = S². To match the two actions on the boundary, it is easy to see that the gluing map

f : S²´ S³ ® S²´ S³

may be written as f (x, y) = (y ^. x, y), where y Î S³ = SU(2) acts on x factoring through the natural SO(3)-action. It is an easy exercise to show that M = .

LEMMA 2.2. Let M be a simply connected compact 6-manifold of positive sectional curvature. If M admits an isometric SO(3)-action such that the fixed point set Fix(M, SO(3)) is not empty, then M is diffeomorphic to S⁶.

PROOF. First we claim that the fixed point set X = Fix(M, SO(3)) must have positive dimension.

In fact, if x Î X is an isolated fixed point, the local isotropy representation of SO(3) gives a linear action on S⁵. But a linear action of SO(3) must have all principal orbit dimension 2, and so the action must have a 2-sphere as the fixed point set (cf. Bredon 1972 exercise in Chap. 5).

By the local isotropy representation we also know that X has dimension 3 and so the principal orbits are of dimension 2. Thus the orbit space is 4-dimensional with boundary X. Hence

M = E(v ) È _¶ SO(3)´ _HD⁴

where E(v ) is a disk bundle of dimension 3 over the fixed point set X, and H is a circle subgroup of SO(3) (cf. Grove and Searle 1994). Clearly, X is a spherical form (cf. Hamilton 1982), since it is totally geodesic. In order that the boundary of the disk bundle matches the boundary of SO(3) ´_HD⁴ the core X must be simply connected. By a straightforward argument one knows that M is a homotopy sphere and therefore M = S⁶.

The q-extent xt_q(X), q³ 2, of a compact metric space (X, d ) is, by definition, given by the following formula:

Recall that the 3-extent (Grove and Markvosen 1995).

LEMMA 2.3. In the orbit space M^* there are at most three isolated singular orbits.

PROOF. We prove by contradiction.

Assume that there are at least 4 isolated singular orbits. Let A₁, A₂, ^... , A₄ be the 4 isolated singular orbits in M^*.

For any i ¹ j, let

C_ij = {g : [0, l_ij] ® M | g is a minimizing normal geodesic from A_i to A_j}.

Let . Since the curvature of M^* is positive (cf. Lemma 1.5), by the Toponogov's comparison theorem it follows that, for each triple (i, j, k) of distinct integers in [1, 4],

a_ijk + a_jik + a_ikj > p

Summing over all possible choices of the triple (i, j, k) we know that

On the other hand, for each A_i the isotropy group H_i

S¹, note that the space of directions at A_i, , where is the unit sphere in the normal slice of the isolated circle orbit A_i in M, which is S³(1). Therefore, åA_i (X) is the quotient of S². Therefore

A contradiction.

PROOF OF THEOREM B. By Lemma 2.1 we may assume that Fix(M, SU(2)) is empty.

By the Synge theorem the center Ì SU(2) must act on M with non empty fixed point (since it preserves the orientation). Consider the fixed point set Ì M. Clearly SU(2) acts on . If it has dimension 4, it must be or S⁴ (cf. Hsiang and Kleiner 1989). But any action of SU(2) on S⁴ or must have fixed point. A contradiction. Therefore has dimension 2 and consists of the union of some isolated singular orbits of dimension 2. Around every singular orbit, there is a regular neighborhood of the form SU(2) ´_HD⁴ where H is the isotropy group of the singular orbit (a 1-dimensional subgroup of SU(2)). In the Alexandrov space M^* = M/SU(2), it gives a neighborhood of the singular orbit and homeomorphic to D⁴/H, a cone over S³/H. Note that the diameter of S³/H is at most . By Lemma 2.3 above we know that there are at most three such isolated singular orbits.

Observe that the orbit space M^* must be a simply connected 3-manifold (therefore M^* is a homotopy 3-sphere) with at most three marked singular points. This shows that M may be re-constructed by gluing at most three handles SU(2)´ H_iD⁴ (with H_i the isotropy groups of the singular orbits) to SU(2) ´ (S³ - È_i int D³_i ), where D³_i , i = 1, 2, 3 are disjoint small disks neighborhood around the singular orbits in M^*. It is easy to check that the third Betti number b₃(M) = 0 and, the Euler characteristic c (M) = 2, 4 or 6, according to 1, 2 or 3 singular orbits.

If c (M) = 2 then M is homotopy equivalent to S⁶ and so M = S⁶ (cf. Wall 1966).

If c (M) = 4 then M = SU(2)´Hi D⁴È_¶ SU(2)´_H2D⁴, where H_i, i = 1, 2 are circle subgroups of SU(2). Since SU(2)´ H_i¶ D⁴ = S²´ S³ we get M = S²´ S⁴.

If c (M) = 6, we now prove that M = SU(3)/T² or M = SU(3)//T².

Let X = S³ - ȳ_i=1 int D³_i . By the above M = SU(2) ´ Xȳ_i=1SU(2)´ H_iD⁴, where f = f₁È f₂È f₃ is the gluing diffeomorphism of ȳ_i=1SU(2)´ H_i¶ D⁴. Since f is SU(2)-equivariant, each component f_i may be identified with the diffeomorphism

f_i : S²´ S³® S²´ S³,(x, y)® (q_i (y) ^. x, y)

where q_iÎ p₃ (SO(3)) is a generator and ^. indicates the standard action of SO(3) on S². Observe that the topology of M depends only on the isotopy class of the diffeomorphism f.

Fix a generator t Î p₃ (SO(3)). Clearly q_i = ± t . For simplicity we let (+ , + , +) denote the manifold M (as well as the map f) with f = (t ,t ,t ). Similarly, (- , - , -) denotes M with f = (- t , - t , - t ), etc.

Let h : S²´ S³® S²´ S³ be a diffeomorphism given by h(x, y) = (x, y^-1), where S³ = SU(2) is considered as a Lie group. Note that the composition of foh will reverse the signs, e.g. foh = (- , - , -) if f = (+ , + , +) and so the manifolds (+ , + , +) and (- , - , -) are the same up to diffeomorphism. Therefore there are at most two different diffeomorphism types (+ , + , +) and (+ , - , -).

On the other hand, as we noted in the beginning of the section, SU(3)/T² and SU(3)//T² both have effective isometric actions by SU(2). The induced actions of SU(2) on the manifolds must have only isolated singular orbits (if not, the manifold must be S⁶ or CP³ by the above argument).

Recall that SU(3)/T² and SU(3)//T² have different homotopy types (cf. Eschenberg 1992). Therefore (+ , + , +) and (+ , - , -) cannot be diffeomorphic. This proves the desired result.

PROOF OF THEOREM C. By Lemma 2.2 we may assume that the fixed point set is empty.

If the principal orbit has dimension 2, since principal orbits are open dense in M by the principal orbit theorem (cf. Bredon 1972), for the principal isotropy group S¹ or O(2), the fixed point set of this isotropy group must be of dimension 4. Therefore M = S⁶ or (cf. Grove and Searle 1994). If SO(3)/H is a principal orbit type, where H Ì SO(3) is a non-trivial finite group. Then the fixed point set M^H is of dimension at least 3 for the same reason as above. Note that the fixed point set must have even codimension since H preserves the orientation (cf. Kobayashi 1972). Therefore M^H has dimension 4. By Theorem 1.5 we may conclude that M = S⁶ or (cf. Fang and Rong 2001).

Now we assume that the principal orbit type is SO(3).

If there is a circle subgroup S¹Ì SO(3) with fixed point set MS¹ of positive dimension, (of course it is not empty by the Synge theorem), then dim MS¹ = 2. Otherwise, M = S⁶ or (cf. Grove and Searle 1994). This is impossible, since any effective SO(3) action on S⁶ or cannot have circle (or O(2)) isotropy group with 4-dimensional fixed point set.

Consider the minimal SO(3)-invariant submanifold containing MS¹. It must have dimension 4 since the orbits starting from points in MS¹ are all of dimension 2. In the orbit space M^* = M/SO(3), the quotient SO(3) ^. MS¹/SO(3) = MS¹ is a boundary component. Let p : M® M^* be the quotient map. By Grove and Searle (1994) again M^* is homeomorphic to D³ and it contains at most one additional singular (or exceptional) orbit outside MS¹ (cf. Grove and Searle 1997). Moreover,

M = D(v ) È_¶ SO(3)´ _HD^{3 + dim H}

where D(v ) is a disk bundle of dimension 2 over the minimal submanifold p^-1 (MS¹), and H is the isotropy group of the unique orbit (it may be principal orbit).

Note that p^-1 (MS¹) = S²´ S². The SO(3) action on this S²´ S² is equivalent to the product action of a trivial SO(3) action on S² and the standard SO(3) on S² (with just one orbit).

If H is a finite group, then H = {1} and so M = . Indeed, in this case the action is equivalent to the one defined by: A ^. [z₀, z₁, z₂, z₃] = [A(z₀, z₁), A(z₂, z₃)], where [z₀, z₁, z₂, z₃] is the homogeneous coordinate and A Î SU(2)/ = SO(3).

We now prove that dim H = 0. In fact, since v is a SO(3)-equivariant oriented 2-bundle over S²´ S². Using standard algebraic topology we know that the Euler class e(v ) has to be even. If H has positive dimension, it must be S¹ or O(2). Since all other orbits are principal, H must be S¹ and so SO(3) ´_HD⁴ = S²´ D⁴. This is impossible since e(v ) = 0(mod 2) implies that boundary SO(3) ´_H¶ D⁴ has fundamental group at least .

Finally we consider the case of MS¹ is zero dimensional for all circle subgroup in SO(3). In other words, singular orbits are isolated. By Lemma 2.3 once again we know that there are at most 3 isolated singular orbits. As in the proof of Theorem B, the possible values of c (M) = 2, 4, 6. In the case c (M) = 4 one has M = SO(3)´Hi D⁴È_¶ SO(3)´_H2D⁴ = S²´ S⁴, up to diffeomorphism. We claim that c (M) = 2 and 6 are impossible. This will complete the proof.

Let X = M^* - È^l_i=1 int D³_i with l the number of singular orbits (here l = 1 or 3), where D³_i is as in Theorem A. Note that M^* is again a homotopy 3-sphere. Since all orbits in X is of type SO(3), the principal SO(3)-action on p^-1 (X) corresponds to a unique classifying map f : X® BSO(3) for the action. Note that f restricts on each piece of the boundary (a 2-sphere) of X gives the classifying map of the free SO(3) action on SO(3)´ H_iS³ = S²´ S³, which is not null homotopic in p₂ (BSO(3)) = . Clearly, the sum of the l boundary pieces must be zero homologous (since the map f is a cycle with boundary the sum). Therefore l must be even. This proves that there are only 2 singular orbits. This proves the desired result.

3. ACKNOWLEDGMENTS

This paper is written based on the author's talks in Brazilian Colloquium in IMPA and in the conference "Foliation and Geometry 2001'' in PUC, Brazil. Supported by CNPq of Brazil, NSFC Grant 19741002, RFDP and Qiu-Shi Foundation of China.

RESUMO

Seja M uma 6-variedade compacta, simplesmente conexa de curvatura sectional positiva. Se a identidade componente do grupo de isometria contém um subgrupo de Lie simples, provamos que M é difeomórfico de uma das 5-variedades listadas no Teorema A.

Palavras-chave: 6-variedades, curvatura positiva, grupos de simetria.

SHANKAR K. 2001b. Private communication.

WILKING B. 2002. In preparation.

E-mail: [e-mail] / [e-mail]

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Publication Dates

• Publication in this collection
  24 Jan 2003
• Date of issue
  Dec 2002

History

• Received
  16 July 2002
• Accepted
  03 Sept 2002