Proceedings of The Fourteenth International Workshop on Diff. Geom. 14(2010) 171-175 On the 5-dimensional Sasaki-Einstein manifold Byung Hak Kim Department of Applied Mathematics, Kyung Hee University, Suwon 446-701, Korea e-mail : bhkim@khu.ac.kr (2010 Mathematics Subject Classification : 53C15, 53C25.) Abstract. In the talk, we will survey the characterizations and constructions of Sasaki- Einstein structure on the fibred Riemannian space. 1 Introduction A Sasakian manifold is a normal contact manifold. In some respects Sasakian manifold may be viewed as odd-dimensional analogue of Kaehler manifolds. Recently there are many papers with respect to the Sasakian geometry, in particular Sasaki-Einstein manifold. Sasaki-Einstein metric on odd-dimensional Riemannian manifold is deeply related to the Kaehler Ricci flat metric, that is Calabi-Yau metric on even-dimensional Riemannian manifold. More precisely Sasaki-Einstein manifold may be defined as an Einstein manifold whose metric cone is Ricci flat and Kaeher, that is Calabi-Yau manifold. Such manifolds provide interesting examples of the string theory [3,4,8]. In fact any complex surface whose metric is Kaehler-Einstein and of positive scalar curvature admits a unique simply connected circle bundle which is canonically Sasaki-Einstein. A classification of Riemannian manifolds admitting real Killing spinors on M correspond to the parallel spinors on C(M) = (R + M, dr 2 + r 2 g) the metric cone on M. In this point of a view, there are many results about Sasaki-Einstein geometry using this cone manifold with the Kaehler structure. Moreover, Sasaki Einstein metric is deeply related to the black hole theory, in particular dimensions 5 or 11. Fibred Riemannian space was first considered by Y. Muto [9] and treated by B. L. Reinhart [12] in the name of foliated Riemannian manifolds. B. O Neill [10] called such a foliation a Riemannian submersion and gave its structure equation. A. Gray [5] have studied pseudo-riemannian almost product manifolds and submersion. In the almost same time S. Ishihara and M. Konishi [6] developed an extensive theory of fibred Riemannian space. M. Ako [1] and T. Okubo [11] studied fibred space Key words and phrases: Sasaki-Einstein manifold, Fibred Riemannian space. 171
172 Byung Hak Kim with almost complex or almost Hermitian structure, and B. Watson [14] studied almost contact metric submersions. Y. Tashiro and B. H. Kim [13] have studied fibred Riemannian space with almost Hermitian or almost contact metric structure. In this point of a view, we will give a talk with respect to the construction of Sasaki-Einstein metric in the fibred Riemannian space and discuss about the characterizations of such spaces. 2 Fibred Riemannian space Let {M, B, G, π} be a fibred Riemannian space, that is M an m-dimensional total space with projectable Riemannian metric G, B an n-dimensional base space, and π : M B a projection with a maximal rank n. The fibre passing through a point q in M is denoted by F (q) or generally F, which is a p-dimensional submanifold of M, where p = m n. The quantities h and L are the components of the second fundamental tensor and normal connection of each fibre respectively. If the horizontal mapping covering curve in M is an isometry (resp. conformal mapping) of fibres, then it is called a fibred Riemannian space with isometric (resp. conformal) fibres. It is well known that a necessary and sufficient condition for M to have isomeric (resp. conformal) fibres is h = 0 (resp. h = λḡ, where ḡ is an induced Riemannian metric on each fibre). The following Theorem is well known [6]. Theorem 2.1. If the components of L and h vanish identically in a fibred Riemannian space, then the fibred space is locally the Riemannian product of the base space and a fibre. By Besse [2], the warped products of two Riemannian manifolds can be considered as a special case of a Riemannian submersion due to the following theorem. Theorem 2.2. Let M = B f 2 F be the warped product of (B, g) and (F, ḡ). Then the projection π : M B onto the first factor is a Riemannian submersion. More over the tensorial invariants of π satisfy ( ) L = 0, h = λḡ and the mean curvature vector is basic. Conversely, the conditions ( ) characterize locally warped products among Riemannain submersions.
On the 5-dimensional Sasaki-Einstein manifold 173 3 Almost contact structure on the fibred Riemannian space There have been various attempts to clarify the relations between almost complex structures and almost contact structures. Using the fact that the structure groups of the tangent bundles of former and of the latter are reduced respectively to the unitary group. For an odd-dimensional manifold M 2n+1, A. Gray [5] defined an almost contact structure as a reduction of the structural group to U(n) 1. In terms of structure tensors we say M 2n+1 has an almost contact structure or sometimes (ϕ, ξ, η)-structure if M admits a tensor field ϕ of type (1, 1), a vector field ξ and a 1-form η satisfying (3.1) ϕ 2 = I + η ξ, η(ξ) = 1. It is well known that (3.1) reduce ϕξ = 0 and η ϕ = 0. If a manifold M 2n+1 with (ϕ, ξ, η)-structure admits a Riemannian metric g such that (3.2) g(ϕx, ϕy ) = g(x, Y ) η(x)η(y ), then we say M 2n+1 has an almost contact metric structure and g is called a compatible metric [3,6]. An almost contact structure (ϕ, ξ, η) on M is normal if the almost complex structure J on M R 1 given by J(X, f d dt ) = (ϕx fξ, η(x) d dt ), f being a C -function on M R 1, is integrable. An almost contact metric manifold (M, g) with (ϕ, ξ, η) is said to be [7,13] (i) contact if Φ = dη (ii) K-contact if Φ = dη and ξ is a Killing vector (iii) Sasakian if Φ = dη and (ϕ, ξ, η) is normal, where Φ(X, Y ) = g(ϕx, Y ). Y.Tashiro and B. H. Kim [13] have studied the fibred almost contact metric space with invariant fibres tangent to the structure vector and deal with various almost contact structure. They considered the fibred Riemannian space M with base space (B, g) with almost complex manifold with almost complex structure J and fibre F with almost contact structure ( ϕ, ξ, η, ḡ). If we put ϕ = J b a E b E a + ϕ β α C β C α, η = η α C α, ξ = ξ α C α, ( g 0 G = 0 ḡ ),
174 Byung Hak Kim then we can easily see that ( ϕ, ξ, η, G) defines an almost contact metric structure on M. Conversely, if there is in M an almost contact structure ( ϕ, ξ, η, G), G and ϕ are projectable and ξ is always vertical, then the structure induces an almost Hermitian structure (J, g) in the base space and almost contact metric structure ( ϕ, ξ, η, ḡ) in each fibre. In this case we have Theorem 3.1. [7] If a fibred almost contact metric structure is Sasakian, then the base space is Kaelerian and each fibre is Sasakian. In this case, each fibre is minimal, and L = J ξ, where J is a almost complex structure on the base space and ξ is a structure vector of the fibre. Theorem 3.2. [7] Let M be fibred Sasakian space with conformal fibres, then M is Sasaki-Einstein if and only if B is Kaeher-Einstein, S = λḡ n η η and K = n(n + 2p + K)/p, where S is a Ricci curvature tensor of the fibre and K is a scalar curvature of the fibre. In this case, each fibre is a totally geodesic submanifold of the total space and S = (α + 2)g, where α = K/m and K is a scalar curvature of the total space. Hence if we consider the 5-dimensional fibred Sasaki-Einstein space with conformal fibres, then we can characterize the geometric structure of the base space and each fibre in two cases, that is n = 4, p = 1 and n = 2, p = 3. Remark 3.3. The Sasakian manifold M can be considered as a cone manifold C(M) with Kaehler structure. Since the warped product is a special case of the fibred Riemanian space [2], we can reduced the properties of the Sasakian structure on M related to C(M) using the fundamental structure equation in the fibred Sasakian space with some conditions. Example 3.4. The Hopf fibration π : S 2n+1 CP n with fibre S 1 is fibred Sasaki- Einstein space with totally geodesic fibre. Obviously, CP n is Kaehler-Einstein. References [1] M. Ako Fibred space with almost complex structure, Kodai Math. Sem. Rep. 24 (1972), 482-505. [2] A. Besse Einstein manifolds, Springer-Verlag, Berlin (1987).
On the 5-dimensional Sasaki-Einstein manifold 175 [3] C.P.Boyer and K.Galicki, On Sasaki-Einstein Geometry, Intl. J. Math. 11 (2000), 873-909 arxiv:math. DG/9811098. [4] Th. Friedrich and I. Kath, Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator, J. Diff. Geom. 29 (1989), 263-273. [5] A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967), 715-738. [6] S. Ishihara and M. Konishi, Differential geometry of fibred spaces, Publ. Study Group of Differential Geometry 7, Tokyo, (1973) [7] B. H. Kim, Fibred Riemannian space with contact structure, Hiroshima Math. J. 18 (1988), 493-508. [8] J. M. Maldacena, The large N limit of superconformal field theories and super gravity, Adv. Theor. Math. Phys. 2 (1998), 231, arxiv:hep-th/9711200. [9] Y. Muto, On some properties of a fibred Riemannian manifold Science reports of Yokohama Natl. Univ., bf 1 (1952), 1-14. [10] B. O Neill, The fundamental equations of submersion, Michigan. Math. J., 13 (1966), 459 469. [11] T. Okubo, Fibred spaces with almost Hermitian metrics whose base space admit almost contact metric structure, Math. Ann., 183 (1969), 290-322. [12] B. L. Reinhart, Foliated manifolds with bundle like metrics, Ann. Math. 69 (1959), 119-132. [13] Y. Tashiro and B. H. Kim, Almost complex and almost contact structures in fibred Riemannian space, Hiroshima Math. J., 18, Ser. A, (1988), 161 188. [14] B. Watson, The differential geometry of two types of almost contact metric submersions, The Math. heritage of C. F. Gauss, (1991), 827 861.