Abstract
A Sasakian structure \(\mathcal{S}\)=(\xi,\eta,\Phi,g) on a manifold Mis called positiveif its basic first Chern class c1(\(\mathcal{F}\) ξ) can be represented by a positive (1,1)-form with respect to its transverse holomorphic CR-structure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This provides us with a new technique for proving the existence of positive Ricci curvature metrics on certain odd dimensional manifolds. As an example we give a completely independent proof of a result of Sha and Yang that for every nonnegative integer kthe 5-manifolds k#(S 2×S 3) admits metrics of positive Ricci curvature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Boyer, C. P. and Galicki, K.: On Sasakian–Einstein geometry, Internat. J. Math. 11(2000), 873–909.
Boyer, C. P. and Galicki, K.: New Einstein metrics in dimension five, math.DG/ 0003174; J. Diff. Geom. 57(2001), 443–463.
Boyer, C. P. and Galicki, K.: Rational homology 5-spheres with positive Ricci curvature, Math. Res. Lett. 9(2002), 521–528.
Boyer, C. P., Galicki, K. and Nakamaye, M.: On the geometry of Sasakian–Einstein 5-manifolds, math.DG/0012047, Math. Ann. 325(2003), 485–524.
Boyer, C. P., Galicki, K. and Nakamaye, M.: Sasakian–Einstein structures on 9#(S 2× S 3), math.DG/0102181; Trans. Amer. Math. Soc. 354(2002), 2983–2996.
Boyer, C. P., Galicki, K. and Nakamaye, M.: Sasakian geometry, homotopy spheres and positive Ricci curvature, Topology, 42(2003), 981–1002.
Braun, V. and Lui, C.-H.: On extremal transitions of Calabi–Yau threefolds and the singularity of the associated 7-space from rolling, hep-th/9801175 v2.
Demailly, J.-P. and Kollár, J.: Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Ann. Sci. Ecole Norm. Sup. Paris 34(2001), 525–556.
El Kacimi-Alaoui, A.: Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compositio Math. 79(1990), 57–106.
El Kacimi-Alaoui, A. and Nicolau, M.: On the topological invariance of the basic cohomology, Math. Ann. 295(1993), 627–634.
Hasegawa, I. And Seino, M.: Some remarks on Sasakian geometry–applications of Myers’ theorem and the canonical affine connection, J. Hokkaido Univ. Education 32(1981), 1–7.
Johnson, J. M. and Kollár, J.: Kähler–Einstein metrics on log del Pezzo surfaces in weighted projective 3-space, Ann. Inst. Fourier 51(1) (2001) 69–79.
Milnor, J.: Singular Points of Complex Hypersurfaces, Ann. of Math. Stud. 61, Princeton Univ. Press, 1968.
Milnor, J. and Orlik, P.: Isolated singularities defined by weighted homogeneous polynomials, Topology 9(1970), 385–393.
Moroianu, S.: Parallel and Killing spinors on Spinc-manifolds, Comm. Math. Phys. 187(1997), 417–427.
Smale, S.: On the structure of 5-manifolds, Ann. Math. 75(1962), 38–46.
Sha, J.-P. and Yang, D.-G.: Positive Ricci curvature on the connectedsums of S n× S m, J. Differential Geom. 33(1991), 127–137.
Tondeur, Ph.: Geometry of Foliations, Monogr. Math., Birkhäuser, Boston, 1997.
Yano, K. and Kon, M.: Structures on Manifolds, Ser. Pure Math. 3, World Scientific, Singapore, 1984.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Boyer, C.P., Galicki, K. & Nakamaye, M. On Positive Sasakian Geometry. Geometriae Dedicata 101, 93–102 (2003). https://doi.org/10.1023/A:1026363529906
Issue Date:
DOI: https://doi.org/10.1023/A:1026363529906