Cohomogeneity One Manifolds and Homogeneous Spaces of Positive Scalar Curvature
aa r X i v : . [ m a t h . DG ] S e p COHOMOGENEITY ONE MANIFOLDS AND HOMOGENEOUS SPACESOF POSITIVE SCALAR CURVATURE
GEORG FRENCK ∗ , FERNANDO GALAZ-GARCÍA ∗† , AND PHILIPP REISER ∗† Abstract.
We characterize cohomogeneity one manifolds and homogeneous spaces with acompact Lie group action admitting an invariant metric with positive scalar curvature. Main Results
Whether a given smooth manifold admits a complete Riemannian metric of positive scalarcurvature is a long-standing problem in Riemannian geometry. For closed (i.e. compact andwithout boundary) simply-connected manifolds of dimension at least 5 this question has beenanswered by Gromov–Lawson [11] and Stolz [32]. For non-simply-connected manifolds, however,the problem is still open in many cases (see, for example, the surveys [34, 35] by Walsh). Undersymmetry assumptions, Lawson and Yau [23] showed that any closed smooth manifold M with a smooth (effective) action of a connected, compact, non-abelian Lie group G supports aninvariant Riemannian metric of positive scalar curvature. Further existence results for manifoldswith circle actions have been obtained by Hanke [20] and Wiemeler [36]. Note that the orbitspace of a smooth effective circle action on an n -manifold, n ≥
1, has dimension n −
1. Thus,one generally thinks of manifolds with circle actions as having high-dimensional orbit spaces. Inthis note we consider the opposite situation, namely, manifolds with compact Lie group actionswhose orbit space is zero- or one-dimensional, and characterize such manifolds admitting positivescalar curvature.Recall that a smooth manifold is a cohomogeneity one manifold if it admits an effective,smooth action of a compact Lie group G and the orbit space M/G of this action is one-dimensional. These manifolds were first studied by Mostert in [26] (see also [10, 25, 28]) andplay an important role in differential geometry (see, for example, [17, 18, 19, 21]). When M isclosed, M/G is homeomorphic to S or [ − , T k denote the k -dimensional torus, K theKlein bottle, and let A be the manifold ( Mb × S ) ∪ ∂ ( S × Mb ), where Mb denotes the Möbiusband. It follows from [28] (or from Theorem A below) that K × S , A , and T are the onlyclosed smooth 3-manifolds admitting a flat Riemannian metric with an effective isometric T action of cohomogeneity one. We get the following characterization of closed cohomogeneityone manifolds with positive scalar curvature. Theorem A.
Let M be a closed, connected, cohomogeneity one manifold of dimension n ≥ .Then the following statements are equivalent:(1) M admits a Riemannian metric of positive scalar curvature.(2) M admits a G -invariant Riemannian metric of positive scalar curvature.(3) M is neither diffeomorphic to a torus nor to a product of a torus with one of K or A .(4) The universal cover of M is not diffeomorphic to Euclidean space.(5) M admits no flat Riemannian metric. Date : September 29, 2020.2010
Mathematics Subject Classification.
Key words and phrases. homogeneous space, cohomogeneity one manifold, positive scalar curvature. ∗ The authors acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foun-dation) – 281869850 (RTG 2229). † Received support from the Deutsche Forschungsgemeinschaft grant GA 2050 2-1 within the SPP 2026 “Ge-ometry at Infinity”. n the case of non-compact cohomogeneity one manifolds the situation is slightly different.Note that we still require the group G to be compact and assume that M has no boundary.Here, M/G is homeomorphic to R or [0 , ∞ ) and we obtain the following result, where Mb o denotes the open Möbius band and the symbol “ ≈ ” denotes diffeomorphism between manifolds.Recall that a manifold is open if it is non-compact and has no boundary. Theorem B.
Let M be an open, connected, cohomogeneity one manifold of dimension n ≥ .Then the following statements are equivalent:(1) M admits a complete Riemannian metric of positive scalar curvature.(2) M admits a complete G -invariant Riemannian metric of positive scalar curvature.(3) M is neither diffeomorphic to T n − × R nor to T n − × Mb o .Furthermore, if M/G ≈ R , then the following statements are equivalent to the previous ones:(4) The universal cover of M is not diffeomorphic to Euclidean space.(5) M has no complete flat Riemannian metric. If G is connected, then the two possibilities in statement (3) of Theorems A and B correspondto the two cases M/G ≈ S or [ − , M/G ≈ R or [0 , ∞ ), in thenon-compact case. Furthermore, the proofs of Theorems A and B show that M admits a flatRiemannian metric if and only if G is abelian, the isotropy subgroups of the principal orbits aretrivial, and the isotropy subgroups of the non-principal orbits are isomorphic to Z .In order to add statements (4) and (5) of Theorem B it is necessary to restrict to the case M/G ≈ R . For example, consider R n with the standard action of O( n ) as a cohomogeneity onemanifold with orbit space [0 , ∞ ). The standard Euclidean metric is flat and invariant underthis action. The torpedo metric however (see for example of [33] or [7]) is an O( n )-invariantmetric of uniformly positive scalar curvature for n ≥
3. If
M/G ≈ R and M has a complete G -invariant metric of positive scalar curvature, then M has in fact a metric of uniformly positivescalar curvature, i.e. the scalar curvature is bounded from below by a positive constant. In thecase that M/G ≈ [0 , ∞ ) this does not hold in general (see Remark 4.2).Theorems A and B also give a classification of flat cohomogeneity one manifolds if the quotient M/G is not [0 , ∞ ). We refer to [24] for similar results on flat cohomogeneity one manifolds.By further increasing the symmetry we arrive at the notion of homogeneous spaces. Recallthat a smooth manifold is a homogeneous space if it admits an effective, smooth, transitiveaction of a Lie group G , i.e. if there is an effective smooth Lie group action with only one orbit,(equivalently, with zero-dimensional orbit space). When G is compact, the following theoremcharacterizes compact homogeneous spaces of positive scalar curvature. Theorem C.
Let M be a homogeneous space of dimension n ≥ and assume that G is compact.Then the following statements are equivalent:(1) M admits a Riemannian metric of positive scalar curvature.(2) M admits a G -invariant Riemannian metric of positive scalar curvature.(3) The connected components of M are not diffeomorphic to a torus.(4) The universal cover of each connected component of M is not diffeomorphic to Euclideanspace.(5) M admits no flat Riemannian metric. Note that the statements in Theorem C are exactly the same as in Theorem A, except foritem (3), where the homogeneous and cohomogeneity one situations differ. If the group G isconnected, then the proof of Theorem C shows that M admits a flat Riemannian metric if andonly if G is abelian and the isotropy subgroup is trivial.The equivalence between items (2) and (4) in Theorem C has already been shown by Bérard-Bergery in [2] and does not require the assumption that G is compact. We also refer to [22]for a different proof of this equivalence in the case of connected Lie groups. The equivalence ofitems (3) and (5) is a special case of a theorem of Wolf [38, Theorem I.2.7.1], who classified flathomogeneous spaces. We will prove Theorem C without resorting to these results. ecall that the Bonnet–Myers theorem implies that the fundamental group of a closed Rie-mannian manifold with positive Ricci curvature must be finite. This condition on the funda-mental group is necessary and sufficient for the existence of an invariant Riemannian metricof positive Ricci curvature both on homogeneous spaces for compact Lie groups and closedcohomogeneity one manifolds. Indeed, a homogeneous space for a compact Lie group has aninvariant Riemannian metric of positive Ricci curvature if and only if its fundamental groupis finite (see [27, Proposition 3.4]). Grove and Ziller showed in [19] that the same equivalenceholds for closed cohomogeneity one manifolds.Homogeneous spaces with an invariant metric of positive sectional curvature, where the Liegroup G must necessarily be compact, have been classified (see, for example, [37] and referencestherein). In the cohomogeneity one case, the possible manifolds that may carry invariant Rie-mannian metrics with positive sectional curvature have been classified in the simply-connectedcase (see [17]). These classifications, however, differ fundamentally from each other and thereis no direct analogy as observed in the case of positive scalar or Ricci curvature.Our note is organized as follows. In Section 2 we prove a result on invariant metrics withnon-negative sectional curvature which simplifies the proofs of the main theorems. In Section 3we prove Theorem C. We then prove Theorems A and B in Section 4. Acknowledgements.
The authors would like to thank Christoph Böhm, Jason DeVito, and Mar-tin Kerin for helpful conversations.2.
Preliminary Observations
Let M be a closed, smooth manifold and let G be a compact Lie group acting smoothlyand effectively on M . We refer the reader to [29, 30] for general background on Riemanniangeometry. For basic results on compact Lie groups and actions on Riemannian manifolds,including homogeneous spaces and cohomogeneity one manifolds, we refer the reader to [1]. Wewill use the following deformation result to obtain metrics with positive scalar curvature. Lemma 2.1.
Suppose that M admits a G -invariant metric g with non-negative scalar curvature.If g is not Ricci-flat, then M admits a G -invariant metric of positive scalar curvature.Proof. One uses the Ricci flow to deform g . As M is compact we have existence and uniquenessof the Ricci flow and it is well known that the deformed metrics are still G -invariant since theRicci flow preserves isometries. The statement now follows directly from [4, Proposition 2.18].Alternatively, the lemma also follows from the deformation techniques of Ehrlich [8], which donot use the Ricci flow. (cid:3) Let us first consider metrics with non-negative sectional curvature. The following propositionfollows from well known results in Riemannian geometry.
Proposition 2.2.
Let M be a closed, connected smooth manifold. Suppose that a compact Liegroup G acts smoothly on M and that M has a G -invariant Riemannian metric of non-negativesectional curvature. Then the following statements are equivalent:(1) M admits a G -invariant metric of positive scalar curvature.(2) M admits no flat metric.(3) The universal cover of M is not diffeomorphic to Euclidean space.(4) M is not finitely covered by a torus.Proof. Suppose that M is finitely covered by a torus. Then the universal cover of M is diffeo-morphic to R n and M is flat by Cheeger and Gromoll’s Splitting Theorem [6]. Hence, by thework of Gromov and Lawson on enlargeable manifolds [12, Theorem A and Corollary A], themanifold M does not admit a metric of positive scalar curvature.To conclude the proof, we prove that item (4) implies item (1). Assume that M has no G -invariant metric of positive scalar curvature. Lemma 2.1 implies that M is Ricci-flat and henceflat, as the sectional curvature is non-negative. It then follows from the Bieberbach theoremsthat M is finitely covered by a torus (see e.g. [5, Theorem II.5.3]). (cid:3) arts of Proposition 2.2 still hold if we weaken the assumptions on the curvature. Proposition 2.3.
Let M be a closed, connected smooth manifold. Suppose that a compactLie group G acts smoothly on M and that M has a G -invariant Riemannian metric of non-negative Ricci curvature. Then statements (2), (3) and (4) of Proposition 2.2 are equivalentand statement (1) implies all other statements. The proof goes along the same lines as that of Proposition 2.2, except that we cannot concludethat M is flat if it has no G -invariant metric of positive scalar curvature. Indeed, the conversedoes not hold in general. Consider the K3 surface: it is Ricci-flat and admits no flat metricbecause it is compact and simply connected. However, it does not admit a Riemannian metricof positive scalar curvature because it is spin with non-vanishing ˆ A -genus. Remark . We will apply Proposition 2.2 to homogeneous spaces in order to prove TheoremC. More generally, we can also consider biquotients
G//H . These are quotients of a compact Liegroup G by the action of a closed subgroup H of G × G that acts on G via ( h , h ) · g = h gh − .Biquotients always admit metrics of non-negative sectional curvature that are invariant underthe canonical action of Norm G × G ( H ) /H (see e.g. [31, Section 2]) and admit invariant metrics ofpositive Ricci curvature if and only if their fundamental group is finite (see [31, Theorem A]).Hence Proposition 2.2 directly applies to this class of spaces. To obtain an analog of Theorem Cfor biquotients one would need to show that every flat biquotient is diffeomorphic to a torus;however, the topological classification of flat biquotients is, to the best of our knowledge, stillopen in full generality.Gromov has conjectured that no closed manifold with contractible universal cover admits ametric of positive scalar curvature (see [13, Conjecture B]). Note that, by the Cheeger–Gromollsplitting theorem, if M is a closed manifold with non-negative Ricci curvature and contractibleuniversal cover f M , then f M is isometric to euclidean space. Hence, by Proposition 2.3, Gromov’sconjecture holds with the extra assumption that the manifold admits a metric of non-negativeRicci curvature. 3. Proof of Theorem C
Let G be a compact Lie group and let M be a homogeneous space for G . Then M isdiffeomorphic to G/H , where H ⊆ G is the isotropy group of some given point p ∈ M . Wefix an Ad G -invariant inner product Q on the Lie algebra g of G , which induces a bi-invariantRiemannian metric on G . The Ad G -invariance of Q implies that(3.1) Q ([ X, Y ] , Z ) = Q ( X, [ Y, Z ])for all
X, Y, Z ∈ g . Let p = h ⊥ be the orthogonal complement of the Lie algebra h of H . We canidentify p with T p M and the isotropy action of H on T p M via the differential corresponds to theaction on p via Ad H . Thus, we can restrict Q to p , which induces a G -invariant Riemannianmetric g on M such that the projection G → G/H is a Riemannian submersion. Hence, fororthonormal vectors
X, Y ∈ p , we have(3.2) sec M ( X, Y ) ≥ sec G ( X, Y ) = 14 | [ X, Y ] | . In particular (
M, g ) has non-negative sectional curvature.By Proposition 2.2 we only have to show that statement (3) of Theorem C implies one of theother statements as all connected components of M are diffeomorphic. Now suppose that M admits no metric of positive scalar curvature. Then the metric g is flat, as it is constant and ofnon-negative sectional curvature, so [ p , p ] = 0, by inequality (3.2). Hence, by (3.1), Q ([ p , h ] , p ) = Q ( h , [ p , p ]) = 0 . Furthermore, again by (3.1), we have Q ([ p , h ] , h ) = Q ( p , [ h , h ]) = 0 , s [ h , h ] ⊆ h , so [ p , h ] = 0. This shows that p ⊆ Z ( g ).As G is compact we can decompose g = [ g , g ] ⊕ Z ( g ) . This decomposition is orthogonal with respect to any Ad G -invariant inner product, so [ g , g ] = Z ( g ) ⊥ ⊆ h . Hence H contains the unique connected closed Lie subgroup S with Lie algebra[ g , g ]. Let M o be a connected component of M . Then M o is a homogeneous space and isdiffeomorphic to G o / ( G o ∩ H ), where G o denotes the identity component of G . The subgroup S is normal and closed in G o , hence we can replace G o and G o ∩ H by their quotient by S . Thus, G o is abelian and G o ∩ H is a normal subgroup. As a consequence, the quotient G o / ( G o ∩ H )is a compact abelian Lie group, i.e. a torus. Hence statement (3) implies statement (1). Thisconcludes the proof of Theorem C. (cid:3) Remark . One could replace the last part of the proof of Theorem C by the following shorter,but less elementary argumentation: Suppose M has no metric of positive scalar curvature. Thenby [23] the identity component G o is abelian, hence the connected components of M , which arediffeomorphic to G o / ( G o ∩ H ), are diffeomorphic to a torus.4. Proofs of Theorems A and B
Let M be a connected cohomogeneity one manifold. By the structure results for cohomo-geneity one manifolds (see, for example, [10, Theorem A and Corollary C] or [15, Section 3]),we have one of the following cases:(C1) M/G ≈ S and M → M/G is a fiber bundle where the fiber is a homogeneous space
G/H with H ⊆ G the principal isotropy of the action.(C2) M/G ≈ [ − ,
1] and M can be written as the union of two tubular neighborhoods D ( G/K ± ) of the non-principal orbits G/K ± with isotropy group K ± . These non-principal orbits project down to the endpoints ± ⊂ [ − , D ( G/K ± ) is equivariantly diffeomorphic to a disk bundle G × K ± D ± , where D ± is a disk normal to the orbit G/K ± . The principal orbits are homogeneous spaces G/H and we have H ⊆ K ± ⊆ G . The quotients K ± /H are diffeomorphic to spheres.(N1) M/G ≈ R and M is the product of R and a homogeneous space G/H .(N2)
M/G ≈ [0 , ∞ ) and, by the slice theorem, M is equivariantly diffeomorphic to a diskbundle G × K D , where D is a disc normal to the non-principal orbit G/K over 0 ∈ [0 , ∞ ).The principal orbits, which correspond to points in (0 , ∞ ), are homogeneous spaces G/H and we have H ⊆ K ⊆ G . The quotient K/H is diffeomorphic to a sphere.Grove and Ziller [18] showed that M admits a G -invariant metric of non-negative sectionalcurvature in some cases and conjectured that this holds in general. This is not the case, however,as shown in [16]. Hence, we cannot derive Theorem A from Proposition 2.2. Instead we will usethe fact that M always admits a G -invariant Riemannian metric of non-negative Ricci curvature.Such metrics were constructed by Grove and Ziller in [19]. We will now go through each one ofthe cases (C1)–(N2) above. We begin with the following observation. Lemma 4.1.
In cases (C1) and (N1) the manifold M admits a G -invariant metric of positivescalar curvature if and only if the fiber G/H does.Proof.
We fix an Ad G -invariant inner product Q on g and set p = h ⊥ as in the proof of The-orem C. In case (C1) the bundle M → S can be considered as the mapping torus of a G -equivariant diffeomorphism G/H → G/H induced by right multiplication R a − on G by anelement a ∈ N ( H ) in the normalizer of H (see, for example, [3, Corollary I.4.3]). The inducedmap on the Lie algebra is Ad a which fixes p as Q is Ad G -invariant. Hence R a − induces an isom-etry on G/H with respect to the metric induced by Q . As a consequence, any Ad G -invariantinner product on g extends to all of M in both cases by taking the product with the flat metricon S or R . In particular M has a metric of non-negative sectional curvature and this metric isflat if and only if its restriction to G/H is flat. This corresponds precisely to the cases where and G/H have no metric of positive scalar curvature by [12, Corollary A] and [12, CorollaryB2]. (cid:3)
Proof of Theorem A in case (C1).
Suppose that M has no G -invariant metric of positivescalar curvature. Then the fiber G/H has no metric of positive scalar curvature by Lemma 4.1.Hence the connected components of
G/H are diffeomorphic to a torus by Theorem C. Byrestricting the action to the identity component G o we obtain that M is a principal T n − -bundle over S . Such bundles are necessarily given by the product T n − × S = T n as S hasno higher homotopy groups and T n − is connected. Hence we have shown that each one ofstatements (3) and (4) imply statement (2). To finish the proof, we proceed as follows. Clearly,statement (2) implies statement (1). Now, by Proposition 2.3, each one of statements (1) and(2) imply statements (4) and (5) , where we consider the trivial group action to get that (1)implies (4) and (5). Again, by Proposition 2.3, statements (4) and (5) are equivalent, and eachone of them implies that M is not finitely covered by a torus, which clearly implies statement(3). This shows the equivalence of statements (1)–(5). (cid:3) Proof of Theorem B in case (N1).
Suppose that M has no G -invariant Riemannian metricof positive scalar curvature. Then, as in case (C1), the connected components of the fiber G/H are diffeomorphic to a torus and M is diffeomorphic to T n − × R . Manifolds of this form admita complete flat Riemannian metric, but have no complete Riemannian metric of positive scalarcurvature by [12, Corollary B2]. This shows that statements (1), (2) and (3) are equivalent andthat they are implied by statement (5).If M admits a G -invariant metric of positive scalar curvature, then so does G/H by Lemma 4.1and, by Theorem C, the universal cover of M is not diffeomorphic to R n . Hence statement (2)implies statement (4). Also statement (4) implies statement (5), as all manifolds with a completeRiemannian metric of non-positive sectional curvature are covered by Euclidean space. (cid:3) Proof of Theorem B in case (N2).
Assume that
M/G is diffeomorphic to [0 , ∞ ), i.e. M can be written as a disc bundle G × K D over the non-principal orbit G/K . Recall that theprincipal orbits of the action are diffeomorphic to
G/H and the non-principal orbit of the action,which projects down to 0 ∈ [0 , ∞ ), has isotropy K with H ⊆ K ⊆ G . We equip M with the G -invariant Riemannian metric g of non-negative Ricci curvature constructed in [19]. This metricis given by(4.1) g = dt + f Q | p + f Q | p + f Q | p + Q | m . Here, t parametrizes a horizontal lift of the orbit space [0 , ∞ ), Q is an Ad G -invariant innerproduct on g , and g is the Q -orthogonal sum h ⊕ p ⊕ m such that h ⊕ p = k , where h and k are the Lie algebras of H and K , respectively. The vector spaces p i are orthogonal subspacesof g that span p . The f i are smooth, odd, non-negative real-valued functions depending on0 ≤ t < ∞ with positive derivative at t = 0.The metric g has non-negative Ricci curvature, so it has positive scalar curvature if it haspositive Ricci curvature in at least one direction at every point. The Ricci curvatures of g werecomputed in [19, Proposition 2.10] and, by [19, Proposition 3.2], they are non-negative if f i ≤ nd the following functions are non-negative:ric t = − X i =0 d i F ′′ i F i , ric = (cid:18) d F + d F (cid:19) F − (cid:18) d F ′ F + d F ′ F (cid:19) F ′ F − F ′′ F , ric = d F F + ( d − (cid:16) − F F − F ′ (cid:17) F + d F F − (cid:18) d F ′ F + d F ′ F (cid:19) F ′ F − F ′′ F , ric = d (3 − F F ) + d (3 − F F ) + ( d − − F ′ ) F − (cid:18) d F ′ F + d F ′ F (cid:19) F ′ F − F ′′ F . Here, d i = dim( p i ), F = abcf , F = bcf , and F = cf , with a, b, c given constants satisfying a, b, c ≥ a, b <
1, and which vanish only if the corresponding subspace p i is trivial.In [19] the functions F i are chosen in the following way (see [19, Lemma 3.3]). Set F ( t ) = c sin( c − t ) on [0 , π c ] and let 0 < t < t < π c such that F ( t ) = abc and F ( t ) = bc . Nowset F = F on [0 , t ], F = bc on [ t , π c ], F = F on [0 , t ], and F = c on [ t , π c ]. Thenthe functions are extended constantly on [ π c, ∞ ), i.e. one lets F i ( t ) = F i ( π c ) for t ∈ [ π c, ∞ ).Then ric t ≥
0. One can now verify that the functions ric i are uniformly positive and that thisstill holds after smoothing these functions if the second derivatives are made sufficiently largearound the non-differentiable points. By [19, Proposition 3.2], the metric g obtained in this wayhas non-negative Ricci curvature.For t > π c , the metric g is a product metric and is flat if the principal orbit G/H cor-responding to t is flat. Thus, to ensure that the scalar curvature is positive, we will modifythe functions F i so that, first, the Ricci curvature of g is non-negative and, additionally, thesecond derivative of the F i is strictly negative for t > t = 0. To achieve this we proceed in a similar way as in the preceding paragraph. Fix asmall ε ∈ (0 , t ), set F ( t ) = c sin( c − t ) on [0 , π c − ε ] and extend F on [ π c − ε, ∞ ) by thefunction t c − t + λ , where λ ∈ R is chosen so that F is continuous. Then set F = F on[0 , t − ε ] and F = F on [0 , t − ε ] and extend these functions as above so that they convergeto bc and abc , respectively, as t → ∞ . In a similar fashion as in the case of non-negative Riccicurvature [19, Lemma 3.3], one can now verify that the functions ric i are uniformly positive for ε sufficiently small. By smoothing the functions F i , again as in [19, Lemma 3.3], one obtainssmooth functions f i with strictly negative second derivative and such that the Ricci curvatureof the metric g is non-negative.By [19, Proposition 2.10], the Ricci curvatures of g for T = ∂∂t and A ∈ m are given byRic( T ) = ric t , (4.2) Ric( A ) = X k k [ A, e k ] h k + 14 k [ A, e k ] m k + X i =0 (cid:18) − f i (cid:19) k [ A, e k ] p i k ! . (4.3)Here ( e k ) denotes an orthonormal basis of m . Hence, g has positive scalar curvature if p isnon-trivial or if there are two vectors A, B ∈ m such that [ A, B ] = 0.Now suppose that M has no G -invariant metric of positive scalar curvature. Then p is trivialand [ m , m ] = 0. By an argument similar to the argument in the proof of Theorem C, it followsthat m ⊆ Z ( g ). Hence, we have(4.4) [ g , g ] = Z ( g ) ⊥ ⊆ m ⊥ = k = h . e consider the action of the identity component G o on M . This action has again cohomogeneityone, but the orbit spaces M/G and
M/G o are not necessarily identical. More precisely, we have M/G o ≈ R or [0 , ∞ ). In the first case we can argue exactly as in case (N1).Suppose now that M/G o ≈ M/G ≈ [0 , ∞ ), so we can replace G , K and H by G o , G o ∩ K and G o ∩ H . By (4.4) the unique connected Lie subgroup S with Lie algebra [ g , g ] is contained in H .Hence, by taking the quotient by S , we can assume that G is abelian. Hence, the subgroup H ,which fixes every point in M , is normal in G . Thus, by taking the quotient by H , we can assumethat H is trivial. The Lie algebras h and k are identical, so K/H is zero-dimensional. As
K/H is diffeomorphic to a sphere, it follows that K is isomorphic to Z . The group G is abelian,hence it is a torus T n − = S × · · · × S ⊆ C × · · · × C , where we choose this identification sothat K ∼ = Z is generated by ( − , , . . . , ∈ T n − . Since the normal tangent space to the orbitis one-dimensional, and hence diffeomorphic to R , it follows that M ≈ G × K R ≈ T n − × ( S × Z R ) ≈ T n − × Mb o . Thus, we have shown that statement (3) implies statement (2).The manifold T n − × Mb o admits no complete metric of positive scalar curvature, since, by[12, Corollary B2], the manifold T n − × R , which double-covers T n − × Mb o , admits no completemetric of positive scalar curvature. This concludes the proof of Theorem B in case (N2). (cid:3) Remark . Note that in case (N2) the manifold M does not necessarily admit a Riemannianmetric of uniformly positive scalar curvature if it admits one with positive scalar curvature.Indeed, consider the standard action of S = SO(2) on R . Then the action of T k − × S = T k on M = T k − × R has cohomogeneity one and admits a complete G -invariant Riemannian metricof positive scalar curvature. Nevertheless, M has no complete metric of uniformly positivescalar curvature (see [14, Section 1]). Proof of Theorem A in case (C2).
Finally, assume that
M/G is diffeomorphic to [ − , M can be written as ( G × K − D − ) ∪ ( G × K + D + ). The metric g of non-negative Riccicurvature constructed in [19] is obtained by gluing two metrics of the form (4.1) on the twohalves G × K ± D ± , where the functions f i are constructed as described in the proof of TheoremA in case (N2) such that they are constant near the gluing area.Suppose that M admits no G -invariant metric of positive scalar curvature. By Proposition2.1 the metric g is Ricci-flat. In this case the formulas (4.2) and (4.3) show that p ± = 0 and h ⊥ = m ± ⊆ Z ( g ). We may now conclude the proof as in case (N2). We consider the action ofthe identity component G o on M . If M/G o ≈ S , i.e. M is a fiber bundle over M/G o with fiber G o / ( G o ∩ H ), then we can argue as in case (N2). If M/G o ≈ M/G ≈ [ − , G , K ± and H by G o , G o ∩ K ± and G o ∩ H , respectively, and, as in case (N2), we canassume that G is abelian, H is trivial, and K ± ∼ = Z . We again write G = T n − = S ×· · ·× S sothat K + is generated by ( − , , . . . ,
1) and K − is generated by ( − , , . . . ,
1) or (1 , − , , . . . , K + = K − or not. In the first case, where K + = K − , we have M ≈ ( G × K + D ) ∪ ∂ ( G × K − D ) ≈ ( Mb × T n − ) ∪ ∂ ( Mb × T n − ) ≈ K × T n − . In the second case, where K + = K − and hence n ≥
3, we have M ≈ ( G × K + D ) ∪ ∂ ( G × K − D ) ≈ ( Mb × S × T n − ) ∪ ∂ ( S × Mb × T n − ) ≈ A × T n − . hus, we have shown that statement (3) implies statement (2) and thus (1). The rest of theproof now follows from Proposition 2.3 as in the proof of case (C1) in Theorem A. (cid:3) Remark . In the proof of case (C2) of Theorem A one could alternatively use [23] to showthat G o is abelian. Furthermore, to conclude that M is diffeomorphic to K × T n − or A × T n − ,one could also argue as follows: A closed, smooth n -manifold, n ≥
3, with an effective actionof T n − is equivariantly diffeomorphic to a product T n − × N , where N is a closed, smooth3-manifold with an effective T action (see, for example, [9, Corollary B]). The possible N arelisted in [28, p. 221]. In our case, the hypothesis that M does not admit a metric with positivescalar curvature implies that N must be diffeomorphic to one of T , K × S , or A , and theonly possibilities that yield an interval orbit space are K × S or A . References [1] Marcos M. Alexandrino and Renato G. Bettiol.
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Institut für Algebra und Geometrie, Karlsruher Institut für Technologie (KIT), Ger-many.
E-mail address : [email protected] E-mail address : [email protected] URL : http://frenck.net/Math (Galaz-García) Department of Mathematical Sciences, Durham University, United Kingdom.
E-mail address : [email protected] (Reiser) Institut für Algebra und Geometrie, Karlsruher Institut für Technologie (KIT), Ger-many and Department of Mathematical Sciences, Durham University, United Kingdom.
E-mail address : [email protected] E-mail address : [email protected]@durham.ac.uk