C. Boyer, K. Galicki, B. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine ang. Math. 455 (1994), 183-220.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....SU(3) SU(2) Theta SU(2) Theta SU(2) or SU(3) Theta SU(2) Theta U(1) In case M 2n 1 admits at least 3 Killing spinors, they introduce a 3 Sasakian structure on M 2n 1 . Manifolds of this type as well as new compact examples have recently been investigated by Boyer Galicki Mann, see [11] [12]. 11 3.2 Imaginary Killing spinors Let (M n ; g) be a complete, connected Riemannian spin manifold and assume that there exists an imaginary Killing spinor 2 Gamma(S) on (M n ; g) rX = iX Delta ; 2 IR: Then (M n ; g) is a non compact Einstein space of negative scalar curvature R ....

Boyer, C.P.; Galicki, K.; Mann, B.M.: The geometry and topology of 3-Sasakian manifolds, Preprint 1993

....manifold, and by a result of Gallot ( 9] the cone over a complete manifold is always irreducible or at as Riemannian manifold. Using the O Neill formulas for warped products, it is easy to relate the di erent geometries of a manifold and of its cone in the following way (see for example [1] or [4] for the de nitions) Theorem 5.3 ( 1] Let M be a Riemannian manifold and M the cone over it. Then M is hyperk ahler or has holonomy Spin 7 if and only if M is a 3 Sasakian manifold, or a weak G 2 manifold respectively. There is an explicit natural correspondence between the above ....

....structures on M and M . This directly yields examples of oriented, non simply connected Riemannian manifolds with holonomy Spin 7 and Sp m , as cones over non simply connected weak G 2 manifolds (cf. 7] or [8] for examples) and non simply connected 3 Sasakian manifolds respectively (cf. [4] for examples) Let now M be a regular simply connected 3 Sasakian manifold other than the round sphere (all known examples of such manifolds are homogeneous) It is a classical fact that M is the total space a SO 3 principal bundle over a quaternionic K ahler manifold, such that the three ....

C. P. BOYER, K. GALICKI, B. M. MANN, The geometry and topology of 3{Sasakian manifolds, J. Reine Angew. Math 455 (1994), 183-220.

....orbit, i.e. the orbit of the highest root vector. This metric is an example of Swann s construction of a hyperkahler metric on the H =Z 2 bundle over a quaternionic Kahler manifold of positive scalar curvature ( 23] It actually follows from the results of Swann ( 23] and Boyer et al. [5,6] that this metric is a cone metric. We will give a different proof of this result: Theorem 3. The adjoint orbit of highest root vectors in a simple Lie algebra g c with Kronheimer s hyperkahler metric is isometric to the punctured cone C (S) over a 3 Sasakian homogeneous manifold S. By the ....

....manifold S. By the punctured cone C (M) over a Riemannian manifold (M; g) we mean the product manifold R Theta M with the metric dr 2 r 2 g. For the definition of a 3 Sasaki structure see section 4. The 3 Sasakian homogeneous manifolds were classified by Boyer, Galicki and Mann in [6] (see also section 4) The simply connected ones are in 1 1 correspondence with simple compact groups. For the classical Lie algebras these authors also described explicitly the metric via certain submersion. Our approach is to analyze Kronheimer s metric directly and it allows us to give the ....

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Boyer, C.P., Galicki, K., and Mann, B.M. The geometry and topology of 3- Sasakian manifolds. J. Reine Angew. Math. 455, 183-220 (1994).

....manifolds always arises as coadjoint orbits. We analyse this situation in more detail to classify those of cohomogeneity two. There has been much recent interest in 3 Sasakian manifolds because they provide new examples of compact Einstein manifolds with positive scalar curvature. As shown in [11], there is a close relationship between 3 Sasakian manifolds, hyperKhler structures and quaternionic Khler orbifolds. Using knowledge of cohomogeneities of adjoint orbits and extending the results of Beauville [5] we are able to classify 3 Sasakian manifolds whose group of 3 Sasakian symmetries ....

....(3) T f Gr 2 (R n ) with G = SO(n) 4) T SO(10) U(5) 5) T E 6 = Spin(10) SO(2) 6) T C P(n) T C P(m) with G = SU(n 1) SU(m 1) 9. Three Sasakian Structures Let S be a compact 3 Sasakian manifold with Riemannian metric g. We refer the reader to [11] for a precise de nition, but for our purposes the following characteristic property will be su cient: the manifold N S : S R 0 with the metric dr 2 r 2 g is hyperKhler with an action of Sp(1) trivial on the R 0 factor and satisfying (i) iv) of Proposition 4.2. This gives examples of ....

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C. P. Boyer, K. Galicki, and B. M. Mann, The geometry and topology of 3- Sasakian manifolds, J. reine angew. Math. 455 (1994), 183220.

.... I 2 ) is an almost hypercomplex structure on the vector bundle S, and this is the announced example Now, we will describe two classes of examples of foliations with projectable, transversal quaternionic structure, which come from 3 Sasakian and quaternion Hermitian Weyl geometry, respectively (cf. [5, 6, 17, 18]) A triple ( 1 , 2 , 3 ) of orthonormal Killing vector fields on a (4q 3) dimensional Riemannian manifold (S; g) is said to define a 3 Sasakian structure if their brackets satisfy the identities [ ff ; fi ] 2 fl 6 ( ff; fi; fl) 1; 2; 3) and cyclic permutations) and, ....

....(I 1 = Gamma Phi 1 =E ; I 2 = Gamma Phi 2 =E ; I 3 = Gamma Phi 3 =E ) is an almost hypercomplex structure on the distribution E = T F . Now, we will check that, although not every I 1 ; I 2 ; I 3 is projectable, the vector bundle Q spanned by these structures is projectable (see also [5, 6]) hence, the foliation V has a projectable, transversal quaternionic structure. Let X be a projectable cross section of E i.e. ff ; X] 2 GammaT V for ff = 1; 2; 3. Then ( ffi D ff Phi fi )X = ff ; Phi fi X] r ff ( Phi fi X) Gamma r Phi fi X ff ) r ff Phi ....

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Ch. P. Boyer, K. Galicki, B. Mann, The geometry and topology of 3Sasakian manifolds, J. Reine Angew. Math., 455 (1994), 183-220.

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C. P. Boyer, K. Galicki, and B. M. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine angew. Math. 455 (1994), 183-220.

....new Einstein and SasakianEinstein manifolds. Using a simple and elegant spectral sequence argument employed by Wang and Ziller we are able to compute the cohomology rings of many examples of the joins of Sasakian Einstein manifolds. This will allow us to generalize some of our previous results [BGMR, BGM2]. For example, we have Corollary B: In every odd dimension greater than 5, there are in nitely many distinct homotopy types of simply connected compact Sasakian Einstein manifolds having the same rational cohomology groups. In particular, in each such dimension, there are in nitely many ....

....be two 3 Sasakian manifolds neither of which are spheres. Then l 1 = l 2 = 1; and the smoothness conditions become gcd(m 1 ; m 2 ) 1: Even this is restrictive since orders tend to be large and have many divisors. However, consider the 3 Sasakian manifolds S(1; 1; 2p i 1) discussed in [BGM2]. In these cases m i = p i 1; so if we choose gcd(p 1 1; p 2 1) 1; which is easy to satisfy, we get a smooth join. Notice, however, that S S is never smooth for S non regular. Finally we consider some non regular examples of manifolds that admit continuous families of Sasakian Einstein ....

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C. P. Boyer, K. Galicki, and B. M. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine angew. Math. 455 (1994), 183-220.

....orbits is very special [Sw] Such orbits admit an action of H with the orbit space being a compact quaternionic K ahler orbifold of positive scalar curvature. Another way of expressing this result is to say that the nilpotent orbits are metric cones C(S) on compact 3 Sasakian orbifolds [BGM1] The mimimal nilpotent orbit is easily seen to be a metric cone C(G=K) where G=K is a simply connected 3Sasakian homogeneous space of [BGM1] Quaternionic geometry of the regular maximal nilpotent orbit of sl(3; C ) was investigated by Kobak and Swann in great detail [KS1] This orbit is ....

....positive scalar curvature. Another way of expressing this result is to say that the nilpotent orbits are metric cones C(S) on compact 3 Sasakian orbifolds [BGM1] The mimimal nilpotent orbit is easily seen to be a metric cone C(G=K) where G=K is a simply connected 3Sasakian homogeneous space of [BGM1] Quaternionic geometry of the regular maximal nilpotent orbit of sl(3; C ) was investigated by Kobak and Swann in great detail [KS1] This orbit is 12 dimensional and, in the language of 3 Sasakian geometry, it is a cone N = C(S) on the 3 Sasakian orbifold S = Z 3 nG 2 =Sp(1) Here S is simply ....

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C.P. Boyer, K. Galicki, and B.M. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine angew. Math., 455 (1994), 183-220.

....new Einstein and SasakianEinstein manifolds. Using a simple and elegant spectral sequence argument employed by Wang and Ziller we are able to compute the cohomology rings of many examples of the joins of Sasakian Einstein manifolds. This will allow us to generalize some of our previous results [BGMR, BGM2]. For example, we have Corollary B: In every odd dimension greater than 5, there are infinitely many distinct homotopy types of simply connected compact Sasakian Einstein manifolds having the same rational cohomology groups. In particular, in each such dimension, there are infinitely many ....

....manifolds neither of which are spheres. Then l 1 = l 2 = 1; and the smoothness conditions become gcd(m 1 ; m 2 ) 1: Even this is restrictive since orders tend to be large and have many divisors. However, consider the 3 Sasakian manifolds S(1; Delta Delta Delta ; 1; 2p i 1) discussed in [BGM2]. In these cases m i = p i 1; so if we choose gcd(p 1 1; p 2 1) 1; which is easy to satisfy, we get a smooth join. Notice, however, that S S is never smooth for S non regular. Finally we consider some non regular examples of manifolds that admit continuous families of Sasakian Einstein ....

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C. P. Boyer, K. Galicki, and B. M. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine angew. Math. 455 (1994), 183-220.

....Sp(n 1) that is, the metric g M is hyperkahler. The fact that every 3 Sasakian manifold S is Einstein and has positive scalar curvature 2(2n 1) 4n 3) is due to Kashiwada [Ka] That S admits a second Einstein metric is a consequence of the canonical variation [Bes] and was given in [BGM2]. The notion of a bundle like metric is due to Reinhart [Rei] That each leaf of F 3 is a totally geodesic 3 Sasakian manifold of constant curvature 1 is due to Kuo and Tachibana [KuTach] and that all 3 Sasakian 3 manifolds are precisely the homogeneous spherical space forms is due to Sasaki ....

.... 3 Sasakian manifold of constant curvature 1 is due to Kuo and Tachibana [KuTach] and that all 3 Sasakian 3 manifolds are precisely the homogeneous spherical space forms is due to Sasaki [Sas2] The regular case of (v) is due to Ishihara and Konishi [IKon,I2] whereas the general case appears in [BGM2]. vi) is given in [BGM1,BGM2] and the proof that Z is a projective algebraic variety is given in [BG] vii) is given independently in [Bar] and [BGM2] We now recall some old results about harmonic forms on compact Sasakian manifolds due to Tachibana [Tach] Consider fS; g; g such that dimS = ....

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C.P. Boyer, K. Galicki, and B.M. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine angew. Math., 455 (1994), 183-220.

....arbitrary cohomogeneity. We believe our examples are different. They fiber over compact Kahler Einstein orbifolds of positive scalar curvature but the circle action has the structure of a Seifert fibration. A detailed study of the geometry of our orbifold examples is currently under investigation [BGM]. 6 Recently Wang introduced another interesting construction of Einstein metrics on some principal bundles over products of quaternionic Kahler manifolds [Wan1] In a sense it is a quaternionic analogue of the construction in [WanZi2] Let M i , 1 i m, be quaternionic Kahler manifolds of ....

....the theorem above one can construct many new non homogeneous Einstein metrics of positive scalar curvature in dimension 4k 3; k 1. There are similar orbifold generalizations of the Wang and Ziller s construction of Einstein metrics on torus bundles over products of Kahler Einstein manifolds [BGM]. Our final example discusses applications of our theory to the hyperkahler structure and associated free Sp(1) action on instanton moduli space thus giving a companion theory to the one developed in [BoMa1] We would like to thank Gerardo Hernandez for helpful conversations about 3 Sasakian ....

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C.P. Boyer K. Galicki and B.M. Mann, The geometry and topology of 3-Sasakian manifolds, UNM preprint, May 1993.

....Z p for any prime p in Lemma 1.6 and carry out a similar analysis to obtain mod p obstructions to smoothness. However, it is not surprising that the p = 2 bound is the sharpest, and as we show in the next sections there are no further Betti number bounds. x2. Some 11 Dimensional Examples. In [BGM] we constructed 3 Sasakian manifolds obtained by toral reduction for k = 1 for every n and in [BGMR1] we constructed 3 Sasakian 7 manifolds with arbitrary k: Thus, to show that Theorem A is sharp we need only construct 11 and 15 dimensional manifold examples when k = 2; 3; 4: These last ....

C.P. Boyer, K. Galicki, and B.M. Mann, The Geometry and Topology of 3-Sasakian Manifolds, J. reine angew. Math. 455 (1994), 183-220.

....are the hyperkahler manifolds where each of the three complex structures is actually Kahler. Until recently, examples of hypercomplex manifolds that are not hyperkahler were rare, the simplest ones being the Hopf manifolds S 4n 3 Theta S 1 which are locally conformally hyperkahler. In [BGM2] we gave a new class of compact, locally conformally hyperkahler manifolds by replacing S 4n 3 with any 3 Sasakian manifold. Hernandez [Her] found similar examples involving the quaternionic Heisenberg group. These non hyperkahler examples, however, are not simply connected. In dimension 4 all ....

....common divisor of all the coordinates is one. A basic sequence is said to be coprime if the coordinates are pairwise relatively prime. If p is an integer multiple of a basic sequence and if the triples (p i ; p j ; k) have no common factor for all 1 i j n then p is called k coprime. In [BGM2] we showed that for all coprime sequences p there is a 3 Sasakian manifold S(p) such that the product S(p) Theta S 1 is a hypercomplex manifold. As mentioned above, these examples should be thought of as generalizations of Hopf manifolds as a 3 Sasakian manifold should be thought of as a ....

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C.P. Boyer K. Galicki and B.M. Mann, The geometry and topology of 3-Sasakian Manifolds, J. reine angew. Math., 455 (1994), 183-220.

....metrics for these first odd dimensional examples are only known to exist implicitly. The purpose of this note is to present a new explicit construction of infinite families of (4n Gamma 5) dimensional Einstein manifolds of positive scalar curvature which are not homogeneous. Definition A: [BGM2] Let n 3 and p = p 1 ; p n ) 2 Z n be an n tuple of nondecreasing, pairwise relatively prime, positive integers. Let S(p) be the left right quotient 1991 Mathematics Subject Classification. Primary 53C25. During the preparation of this work all three authors were supported by NSF ....

.... Gamma Gamma Gamma 0 p1 . pn 1 A W I 2 O O B : Here W 2 U(n) and ( B ) 2 U(1) Theta U(n Gamma 2) Equivalently, S(p) is the quotient of the complex Stiefel manifold V C n;2 of 2 frames in C n by the specific free left circle action which depends on p: Theorem B: [BGM2] Let n 3 and p = p 1 ; p n ) 2 Z n be an n tuple of nondecreasing, pairwise relatively prime, positive integers. Then S(p) is a compact, simply connected, 4n Gamma 5) dimensional smooth manifold which admits an Einstein metric g(p) of positive scalar curvature and a compatible ....

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C.P. Boyer K. Galicki and B.M. Mann, The Geometry and Topology of 3-Sasakian Manifolds, University of New Mexico and Max-Planck-Institute preprints, June 1993.

....beginning of our efforts to understand the geometry and topology of 3 Sasakian manifolds. They have led us through the classification of all 3 Sasakian homogeneous spaces and a discovery of a new quotient construction of infinitely many homotopy types of non regular compact 3 Sasakian manifolds [26]. In dimension 7 these examples turned out to be certain Eschenburg bi quotients of U(3) by a 2 torus [40] and [41] We gave a complete analysis of the geometry and topology of such spaces [26] The next important step was the second author s work with Simon Salamon [54] There we noticed that ....

.... a new quotient construction of infinitely many homotopy types of non regular compact 3 Sasakian manifolds [26] In dimension 7 these examples turned out to be certain Eschenburg bi quotients of U(3) by a 2 torus [40] and [41] We gave a complete analysis of the geometry and topology of such spaces [26]. The next important step was the second author s work with Simon Salamon [54] There we noticed that Kuo s theorem about odd Betti numbers of 3 Sasakian manifolds being divisible by 4 missed a crucial point. Because of the isometric SU(2) action, all odd Betti numbers up to the middle dimension ....

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C. P. Boyer, K. Galicki, and B. M. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine angew. Math. 455 (1994), 183-220.

....homotheties we get an induced action of G on M; and by Proposition 3.2 we can identify G with a Lie subgroup of C(M; j) This gives a moment map by restriction, viz. M Gamma Gamma g ; 4:5 = j M Thetaf1g ; j(X ) Such a moment map was noticed previously by Geiges [Gei] and in [BGM] within the context of 3 Sasakian geometry. We wish to consider the special case when the Lie group G is an n 1 dimensional torus T n 1 : Let t n 1 denote the Lie algebra of T n 1 ; and let fe i g n i=0 denote the standard basis for t n 1 R n 1 : Corresponding to each basis element e i ....

....by this and 0 r i 1 a i : The special case where a i = a for all i = 0; n is just the dilated standard n simplex a(r 0 Delta Delta Delta r n ) 1; 0 r i 1 a : 5. A Delzant Theorem for Toric Contact Manifolds of Reeb type We begin by considering contact reduction [BGM, Gei]. Let ( M ; j) be a compact contact manifold with a fixed quasi regular contact form j: Suppose also that a compact Lie group G acts on M preserving the contact form j and let : M Gamma Gamma g denote the corresponding moment map. Then if G acts freely on the zero set Gamma1 ....

C. P. Boyer, K. Galicki, and B. M. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine angew. Math. 455 (1994), 183-220.

....i (S; Z) Omega Q : For example, the rational Gysin sequence Delta Delta Delta Gamma Gamma H p (S; Q) Gamma Gamma H p 2 (S; Q ) Gamma Gamma H p 2 (H(S) Q) Gamma Gamma H p 1 (S; Q) Gamma Gamma Delta Delta Delta applies. Combining this with known results of 3 Sasakian manifolds [BGM1,GS] one finds Proposition 2.15: Let H(S) be a circle bundle over a 3 Sasakian orbifold S of dimension 4n Gamma 1: Then the following relations for the Betti numbers of H(S) hold: 12 (i) b 2n (H(S) 0: ii) b 2n Gamma2 (H(S) b 2n Gamma1 (H(S) b 2n Gamma2 (S) iii) b 2p 1 (H(S) b 2p (S) ....

....acts effectively on the S 4n Gamma1 : The group Sp(n) is precisely the subgroup of the isometry group which commutes with the 3 Sasakian Sp(1) action, i.e. the group of automorphisms preserving the 3 Sasakian structure. Associated to any subgroup G ae Sp(n) there is a 3 Sasakian moment map [BGM1] G : S 4n Gamma1 Gamma Gamma g Omega R 3 ; where g denotes the dual of the Lie algebra g of G: In this paper we shall consider a maximal torus T n ae Sp(n) and its subgroups acting on S 4n Gamma1 . The maximal torus that we choose is that given in terms of its action on H n by u ff ....

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C.P. Boyer, K. Galicki, and B.M. Mann, The Geometry and Topology of 3-Sasakian Manifolds, J. reine angew. Math. 455 (1994), 183-220.

....Moreover, there is a 1 1 correspondence between compact Kahler Einstein Fano contact manifolds up to biholomorphism and compact quaternionic Kahler manifolds of positive scalar curvature up to homothety. It is also well known by now that 3 Sasakian geometry is intimately related to this setup [6,7]. In fact, the twistor space is just the total space of a certain 2 sphere bundle over a quaternionic Kahler manifold, and the 3 Sasakian manifold is the associated principal SU(2) or SO(3) bundle [6,27] Thus, LeBrun s theorem should have a 3 Sasakian version. However, there is an essential ....

....orbifold counterparts. However, since it is 3 Sasakian geometry that interests us most, one of our main results gives the precise correspondence between 3 Sasakian geometry and twistor geometry. This is carried out in section 4. Let (S; g; a ) be a 3 Sasakian manifold of dimension 4n 3: See [6,7] and references therein for full details. Let S 1 a denote the locally free circle action generated by the vector field a : Then S has a one dimensional foliation with compact leaves, and the space of leaves Z; which we call the twistor space of S; is fairly well behaved. In fact, much more ....

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C.P. Boyer, K. Galicki, and B.M. Mann, The geometry and topology of 3-Sasakian Manifolds, J. reine angew. Math. 455 (1994), 183-220.

....the same rational cohomology as S 1 S 2 : Using a simple and elegant spectral sequence argument employed by Wang and Ziller we are able to compute the cohomology rings of many examples of the joins of SasakianEinstein manifolds. This will allow us to generalize some of our previous results [BGMR, BGM2]. For example, we have Corollary B: In every odd dimension greater than 5, there are infinitely many distinct homotopy types of simply connected compact Sasakian Einstein manifolds having the same rational cohomology groups. In particular, in each such dimension, there are infinitely 2 many ....

....manifold. Then S is regular. Moreover, it is an S 1 bundle over a generalized flag manifold G=P: Conversely, given any generalized flag manifold G=P there is a circle bundle : S Gamma Gamma G=P whose total space S is a homogeneous Sasakian Einstein manifold. Proof: As in Proposition 4. 6 of [BGM2], S is regular. By Theorem 2.5 S fibers over a simply connected Fano variety Z with a Kahler Einstein metric of positive scalar curvature. Since the action of K commutes with it sends fibers to fibers, and thus acts transitively on Z: But by Wang s theorem [Akh] Z = G=P for some complex ....

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C. P. Boyer, K. Galicki, and B. M. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine angew. Math. 455 (1994), 183-220.

....nilpotent orbits is very special [Sw] Such orbits admit an action of H with the orbit space being a compact quaternionic Kahler orbifold of positive scalar curvature. Another way of expressing this result is to say that the nilpotent orbits are metric cones C(S) on compact 3 Sasakian orbifolds [BGM1] The mimimal nilpotent orbit is easily seen to be a metric cone C(G=K) where G=K is a simply connected 3Sasakian homogeneous space of [BGM1] Quaternionic geometry of the regular maximal nilpotent orbit of sl(3; C ) was investigated by Kobak and Swann in great detail [KS1] This orbit is ....

....positive scalar curvature. Another way of expressing this result is to say that the nilpotent orbits are metric cones C(S) on compact 3 Sasakian orbifolds [BGM1] The mimimal nilpotent orbit is easily seen to be a metric cone C(G=K) where G=K is a simply connected 3Sasakian homogeneous space of [BGM1] Quaternionic geometry of the regular maximal nilpotent orbit of sl(3; C ) was investigated by Kobak and Swann in great detail [KS1] This orbit is 12 dimensional and, in the language of 3 Sasakian geometry, it is a cone N = C(S) on the 3 Sasakian orbifold S = Z 3 nG 2 =Sp(1) Here S is simply ....

[Article contains additional citation context not shown here]

C.P. Boyer, K. Galicki, and B.M. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine angew. Math., 455 (1994), 183-220.

....are the hyperkahler twisted products of K3 surfaces constructed by Beauville [Bea] Examples of hypercomplex manifolds that are not hyperkahler were very scarce, the simplest ones being the Hopf manifolds S 4n 3 Theta S 1 which are locally conformally hyperkahler. Recently the authors [BGM2] gave a class of new compact locally conformally hyperkahler manifolds by replacing S 4n 3 with any 3 Sasakian manifold. Similar examples involving the quaternionic Heisenberg group were found by Hernandez [Her] None of these examples, however, are simply connected. During the preparation of ....

....[W] admitting complex structures, respectively. Using different methods, Joyce [Joy1] later recovered this [SSTP] classification and developed a theory of homogeneous hypercomplex manifolds which generalizes Wang s [W] result. In this paper, guided by our previous work on 3 Sasakian manifolds [BGM1,BGM2], we prove the existence of uncountably many distinct hypercomplex structures on Stiefel manifolds of complex 2 planes in complex n space. Our main results are as follows. Theorem A: Let n 2 and p = p 1 ; p n ) 2 (R ) n be an n Gammatuple of non zero real numbers. For each such p ....

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C.P. Boyer K. Galicki and B.M. Mann, The geometry and topology of 3-Sasakian Manifolds, J. reine angew. Math., 455 (1994), 183-220.

.... combined with results described herein, implies that given a number 0 there are an infinite number of positive Einstein manifolds that do not admit metrics with sectional curvatures bounded below by : The technique that we use to construct our examples is the 3 Sasakian reduction procedure [BGM2] starting from the standard 3 Sasakian sphere (S 4n Gamma1 ; g can ) Thus, the positive Einstein manifolds that we describe are 3 Sasakian. Our construction is described During the preparation of this work the first three authors were supported by an NSF grant. 1 in the next section and ....

....homology type. It is clear that our examples do not satisfy the necessary conditions that guarentee many of the well known finiteness results (cf. Che] However, one can contrast the examples given here which do not admit metrics of positive sectional curvature with our previous examples [BGM2,BGM3] as well as the Einstein manifolds of [Wa] In those examples one has positive Einstein manifolds with b 2 = 1; and with infinitely many distinct homotopy types. However, many of those examples admit metrics with positive sectional curvature. Furthermore, the manifolds in [Wa] are diffeomorphic to ....

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C.P. Boyer K. Galicki and B.M. Mann, The geometry and topology of 3-Sasakian Manifolds, J. reine angew. Math., 455 (1994), 183-220.

....Einstein metrics. In dimensions bigger than four almost nothing seems to be known in general. Yet, Einstein metrics on compact manifolds are relatively rare and they usually appear as part of additional geometric structure which makes their study tractable. In recent years the first three authors [BGM1,BGM2] have studied a class of Riemannian manifolds known as 3 Sasakian manifolds which have proven to be a remarkable source of compact Einstein manifolds of positive scalar curvature. In view of this work several seemingly unrelated questions regarding the possible breakdown of finiteness and Betti ....

....scalar curvature. Another result relating to the breakdown of finiteness comes from a theorem of Anderson [An] which implies there is finite number of diffeomorphism types of Einstein manifolds of positive scalar curvature with a lower bound on the injectivity radius. Thus using our examples in [BGM2,BGM3] or those of Theorem A give: Corollary C: There are infinitely many 3 Sasakian 7 manifolds with arbitrarily small injectivity radius. 2 It is interesting to compare the examples of Theorem A with our previous examples [BGM2] where in particular we constructed infinitely many 3 Sasakian ....

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C.P. Boyer K. Galicki and B.M. Mann, The geometry and topology of 3-Sasakian Manifolds, J. reine angew. Math., 455 (1994), 183-220.

.... that there are inhomogeneous Kahler Einstein metrics on the del Pezzo surfaces C P 2 #k( GammaC P 2 ) for 3 k 8: When M is odd dimensional the only explicit examples of inhomogeneous Einstein manifolds of positive scalar curvature with arbitrary cohomogeneity were obtained by the authors in [BGM1, BGM2, BGM3]. All of these examples are 3 Sasakian manifolds. A different construction of odd dimensional inhomogeneous Einstein spaces is due to Wang and Ziller [WZ1, WZ2] They obtained Einstein metrics of positive scalar curvature on certain torus bundles over products of Kahler Einstein spaces. This ....

....torus bundle. A similar construction of Einstein metrics on certain principal RP 3 bundles over products of quaternionic Kahler manifolds is due to Wang [W2] Following Eschenberg [E1] one says that M is strongly inhomogeneous if M is not homotopy equivalent to any homogeneous space G=H: In [BGM2, BGM3] we constructed inhomogeneous Einstein metrics of positive scalar curvature on compact simply connected 3 Sasakian manifolds (S(p) g(p) in dimension 4n Gamma 5 for all n 3: The metrics obtained there are all inhomogeneous as can be easily seen from the geometry of the construction. The ....

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C.P. Boyer, K. Galicki, and B.M. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine angew. Math. 455 (1994), 183-220.

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C. Boyer, K. Galicki, B. Mann, The geometry and topology of 3-Sasakian manifolds, J. reine ang. Math. 455 (1994), 183-220.

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