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2024 | OriginalPaper | Chapter

Some Topics in Sasakian Geometry, a Survey

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Abstract

In the seminal book Sasakian Geometry by C. Boyer and K. Galicki, the authors formulated a research program for studying topological properties and answering questions about the existence of Sasakian structures. We survey recent progress in this topic.

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Literature
1.
2.
go back to reference Barth, W., Hulek, K., Peters, C. and Van de Ven, A.: Compact Complex Surfaces, Springer, 2004 Barth, W., Hulek, K., Peters, C. and Van de Ven, A.: Compact Complex Surfaces, Springer, 2004
3.
go back to reference Bazzoni, G., Biswas, I., Fernández, M., Muñoz, V. and Tralle, A.: Homotopic properties of Kähler orbifolds, in: S.G. Chiossi et. al (eds.) Special Metrics and Group Actions in Geometry, Springer INdAM Series 23, 2017, pp. 23–57 Bazzoni, G., Biswas, I., Fernández, M., Muñoz, V. and Tralle, A.: Homotopic properties of Kähler orbifolds, in: S.G. Chiossi et. al (eds.) Special Metrics and Group Actions in Geometry, Springer INdAM Series 23, 2017, pp. 23–57
4.
go back to reference Biswas, I., M. Fernández, V. Muñoz, Tralle, A.: On formality of Sasakian manifolds, J. Topology 9 (2016), 161–180 Biswas, I., M. Fernández, V. Muñoz, Tralle, A.: On formality of Sasakian manifolds, J. Topology 9 (2016), 161–180
5.
go back to reference Boyer, C., and Galicki, K.: Sasakian Geometry, Oxford, 2007 Boyer, C., and Galicki, K.: Sasakian Geometry, Oxford, 2007
7.
go back to reference Boyer, C., Galicki, K. and Matzeu, P.: On \(\eta \)-Einstein Sasakian geometry, Commun. Math. Phys. 262 (2006), 177–208MathSciNetCrossRef Boyer, C., Galicki, K. and Matzeu, P.: On \(\eta \)-Einstein Sasakian geometry, Commun. Math. Phys. 262 (2006), 177–208MathSciNetCrossRef
8.
go back to reference Boyer, C., Galicki, K. and Mann, B.: The geometry and topology of 3-Sasakian manifolds, J. Reine Angew. Math. 455(1994), 183–220MathSciNet Boyer, C., Galicki, K. and Mann, B.: The geometry and topology of 3-Sasakian manifolds, J. Reine Angew. Math. 455(1994), 183–220MathSciNet
9.
10.
12.
go back to reference Cappelletti-Montano, B., de Nicola, A. and Yudin, I.: Hard Lefschetz theorem for Sasakian manifolds, J. Diff. Geom. 101 (2015), 47–66MathSciNet Cappelletti-Montano, B., de Nicola, A. and Yudin, I.: Hard Lefschetz theorem for Sasakian manifolds, J. Diff. Geom. 101 (2015), 47–66MathSciNet
13.
go back to reference Cappelletti-Montano, B., de Nicola, A., Marrero, J.C. and Yudin, I.: Examples of compact K-contact manifolds with no Sasakian metric, Intenat. J. Geom. Methods Phys. 11 (2014), art. no. 1460028 Cappelletti-Montano, B., de Nicola, A., Marrero, J.C. and Yudin, I.: Examples of compact K-contact manifolds with no Sasakian metric, Intenat. J. Geom. Methods Phys. 11 (2014), art. no. 1460028
14.
go back to reference Cavalcanti, G.: The Lefschetz property, formality and blowing-up in symplectic geometry, Trans. Amer. Math. Soc. 359 (2007), 333–348MathSciNetCrossRef Cavalcanti, G.: The Lefschetz property, formality and blowing-up in symplectic geometry, Trans. Amer. Math. Soc. 359 (2007), 333–348MathSciNetCrossRef
15.
go back to reference Cordero, L.A., Fernández, M. and de León, M.: Examples of compact almost contact manifolds admitting Sasakian and no cosymplectic structures, Atti Sem. Mat. Fis. Univ. Modena 34 (1985), 43–54MathSciNet Cordero, L.A., Fernández, M. and de León, M.: Examples of compact almost contact manifolds admitting Sasakian and no cosymplectic structures, Atti Sem. Mat. Fis. Univ. Modena 34 (1985), 43–54MathSciNet
16.
go back to reference Cañas, A., Muñoz, V., Schütt, M. and Tralle, A.: Quasi-regular Sasakian and K-contact structures on Smale-Barden manifolds, Rev. Matem. Iberoam. 38 (2022), 1029–1050MathSciNetCrossRef Cañas, A., Muñoz, V., Schütt, M. and Tralle, A.: Quasi-regular Sasakian and K-contact structures on Smale-Barden manifolds, Rev. Matem. Iberoam. 38 (2022), 1029–1050MathSciNetCrossRef
17.
go back to reference Cañas, A., Muñoz, V., Rojo, J. and Viruel, A.: A K-contact simply connected 5-manifold with no semi-regular Sasakian structure, Publ. Math. 65 (2021), 615–651MathSciNetCrossRef Cañas, A., Muñoz, V., Rojo, J. and Viruel, A.: A K-contact simply connected 5-manifold with no semi-regular Sasakian structure, Publ. Math. 65 (2021), 615–651MathSciNetCrossRef
18.
go back to reference Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D.: Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245–274MathSciNetCrossRef Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D.: Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245–274MathSciNetCrossRef
19.
go back to reference Donaldson, S.: Symplectic submanifolds and almost-complex geometry, J. Diff. Geom. 44 (1996) 666–705MathSciNet Donaldson, S.: Symplectic submanifolds and almost-complex geometry, J. Diff. Geom. 44 (1996) 666–705MathSciNet
20.
go back to reference El Kacimi Alaoui, A.: Operateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compositio Math. 73 (1990), 57–106MathSciNet El Kacimi Alaoui, A.: Operateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compositio Math. 73 (1990), 57–106MathSciNet
21.
go back to reference Felix, Y., Oprea, J. and Tanré, D.: Algebraic Models in Geometry, Oxford Univ. Press, 2008CrossRef Felix, Y., Oprea, J. and Tanré, D.: Algebraic Models in Geometry, Oxford Univ. Press, 2008CrossRef
22.
go back to reference Felix, Y., Thomas, J.-C. and Halperin, S.: Rational homotopy theory, Springer, Berlin, 2002 Felix, Y., Thomas, J.-C. and Halperin, S.: Rational homotopy theory, Springer, Berlin, 2002
23.
go back to reference Gómez, R.: A note on Smale manifolds and Lorentzian Sasaki-Einstein geometry, Bull. Math. Soc. Sci. Roumainie 59 (2016), 151–158 Gómez, R.: A note on Smale manifolds and Lorentzian Sasaki-Einstein geometry, Bull. Math. Soc. Sci. Roumainie 59 (2016), 151–158
24.
go back to reference Gompf, R.: A new construction of symplectic manifolds, Annals Math. 142 (1995), 527–595 Gompf, R.: A new construction of symplectic manifolds, Annals Math. 142 (1995), 527–595
25.
go back to reference Gompf, R. and Stipsicz, A.: \(4\)-Manifolds and Kirby Calculus, AMS, 2004 Gompf, R. and Stipsicz, A.: \(4\)-Manifolds and Kirby Calculus, AMS, 2004
26.
go back to reference Hajduk, B.and Tralle, A.: On simply connected compact \(K\)-contact non-Sasakian manifolds, J. Fixed Point Theory Appl. 16 (2014), 229–241MathSciNetCrossRef Hajduk, B.and Tralle, A.: On simply connected compact \(K\)-contact non-Sasakian manifolds, J. Fixed Point Theory Appl. 16 (2014), 229–241MathSciNetCrossRef
27.
go back to reference Halperin, S.: Lectures on minimal models, Mém. Soc. Math. France 230, 1983 Halperin, S.: Lectures on minimal models, Mém. Soc. Math. France 230, 1983
28.
go back to reference Hartshorne, R.: Algebraic Geometry, Springer, 1977 Hartshorne, R.: Algebraic Geometry, Springer, 1977
29.
30.
33.
go back to reference McDuff, D. and Salamon, D.: Introduction to Symplectic Topology, Oxford Univ. Press, 1998 McDuff, D. and Salamon, D.: Introduction to Symplectic Topology, Oxford Univ. Press, 1998
34.
go back to reference Muñoz, V.: Gompf connected sum for orbifolds and K-contact Smale-Barden manifolds, Forum Mathematicum, 34 (2022), 197–223MathSciNetCrossRef Muñoz, V.: Gompf connected sum for orbifolds and K-contact Smale-Barden manifolds, Forum Mathematicum, 34 (2022), 197–223MathSciNetCrossRef
36.
go back to reference Muñoz, V., Rojo, J.A. and Tralle, A.: Homology Smale-Barden manifolds with K-contact and Sasakian structures, Internat. Math. Res. Notices IMRN, Vol. 2020, (2020), 7397–7432MathSciNetCrossRef Muñoz, V., Rojo, J.A. and Tralle, A.: Homology Smale-Barden manifolds with K-contact and Sasakian structures, Internat. Math. Res. Notices IMRN, Vol. 2020, (2020), 7397–7432MathSciNetCrossRef
37.
go back to reference Muñoz, V. and Tralle, A.: On the classification of Smale-Barden manifolds with Sasakian structures, Comm. Contemporary Math., 24 (2022), art. no. 2150077 Muñoz, V. and Tralle, A.: On the classification of Smale-Barden manifolds with Sasakian structures, Comm. Contemporary Math., 24 (2022), art. no. 2150077
38.
go back to reference Muñoz, V. and Tralle, A.: Simply connected K-contact and Sasakian manifolds in dimension 7, Math. Z. 281 (2015), 457–470MathSciNetCrossRef Muñoz, V. and Tralle, A.: Simply connected K-contact and Sasakian manifolds in dimension 7, Math. Z. 281 (2015), 457–470MathSciNetCrossRef
39.
go back to reference Muñoz, V., Schütt, M. and Tralle, A.: Negative Sasakian structures on simply connected 5-manifolds, Math. Res. Lett. 29 (2022), 1827–1857MathSciNetCrossRef Muñoz, V., Schütt, M. and Tralle, A.: Negative Sasakian structures on simply connected 5-manifolds, Math. Res. Lett. 29 (2022), 1827–1857MathSciNetCrossRef
40.
go back to reference Nori, M.: Zariski’s conjecture and related problems, Annales Sci. l’E.N.S. 4e série, 16 (1983), 305–344 Nori, M.: Zariski’s conjecture and related problems, Annales Sci. l’E.N.S. 4e série, 16 (1983), 305–344
41.
go back to reference Patra, D.S. and Rovenski, V.: On the rigidity of the Sasakian structure and characterization of cosymplectic manifolds, Differential Geometry and its Applications, 90 (2023) 102043MathSciNetCrossRef Patra, D.S. and Rovenski, V.: On the rigidity of the Sasakian structure and characterization of cosymplectic manifolds, Differential Geometry and its Applications, 90 (2023) 102043MathSciNetCrossRef
42.
go back to reference Park, J. and Won, J.: Simply connected Sasaki-Einstein rational homology spheres, Duke Math. J. 170 (2021), 1085–1112MathSciNetCrossRef Park, J. and Won, J.: Simply connected Sasaki-Einstein rational homology spheres, Duke Math. J. 170 (2021), 1085–1112MathSciNetCrossRef
43.
go back to reference Rukimbira, R.: Chern-Hamilton conjecture and K-contactness, Houston J. Math. 21 (1995), 709–718MathSciNet Rukimbira, R.: Chern-Hamilton conjecture and K-contactness, Houston J. Math. 21 (1995), 709–718MathSciNet
44.
go back to reference Sasaki, S. and Hatakeyama, Y.: On differentiable manifolds with certain structures which are closely related to almost contact structure, II, Tôhoku Math. J. 13 (1961), 281–294MathSciNetCrossRef Sasaki, S. and Hatakeyama, Y.: On differentiable manifolds with certain structures which are closely related to almost contact structure, II, Tôhoku Math. J. 13 (1961), 281–294MathSciNetCrossRef
45.
46.
go back to reference Tievsky, A.: Analogues of Kähler geometry on Sasakian manifolds, Ph.D. Thesis, MIT, 2008 Tievsky, A.: Analogues of Kähler geometry on Sasakian manifolds, Ph.D. Thesis, MIT, 2008
47.
go back to reference Tralle, A. and Oprea, J.: Symplectic manifolds with no Kähler structure, Springer, Berlin, 1997CrossRef Tralle, A. and Oprea, J.: Symplectic manifolds with no Kähler structure, Springer, Berlin, 1997CrossRef
48.
go back to reference Wang, Z. and Zafran, D.: A remark on the hard Lefschetz theorem for Kähler orbifolds, Proc. Amer. Math. Soc. 137 (2009), 2497–2501MathSciNetCrossRef Wang, Z. and Zafran, D.: A remark on the hard Lefschetz theorem for Kähler orbifolds, Proc. Amer. Math. Soc. 137 (2009), 2497–2501MathSciNetCrossRef
Metadata
Title
Some Topics in Sasakian Geometry, a Survey
Author
Aleksy Tralle
Copyright Year
2024
DOI
https://doi.org/10.1007/978-3-031-50586-7_1

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