Abstract
We discuss local Sasakian immersion of Sasaki–Ricci solitons (SRS) into fiber products of homogeneous Sasakian manifolds. In particular, we prove that SRS locally induced by a large class of fiber products of homogeneous Sasakian manifolds are, in fact, \(\eta \)-Einstein. The results are stronger for immersions into Sasakian space forms. Moreover, we show an example of a Kähler–Ricci soliton on \(\mathbb C^n\) which admits no local holomorphic isometry into products of homogeneous bounded domains with flat Kähler manifolds and generalized flag manifolds.
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References
Futaki, A., Ono, H., Wang, G.: Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. J. Differ. Geom. 83(3), 585–635 (2009)
Smoczyk, K., Wang, G., Zhang, Y.: The Sasaki-Ricci flow. Int. J. Math. 21(7), 951–969 (2010)
Tristan, C.: Collins, Stability and convergence of the Sasaki-Ricci flow. J. Reine Angew. Math. 716, 1–27 (2016)
Collins, T.C., Jacob, A.: On the convergence of the Sasaki-Ricci flow, Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong. Contemp. Math., vol. 644, Am. Math. Soc., Providence, RI, pp. 11–21 (2015)
Petrecca, D.: On sasaki-ricci solitons and their deformations. Adv. Geom. 16(1), 57–70 (2016)
Tadano, H.: Gap theorems for compact gradient Sasaki-Ricci solitons. Int. J. Math. 26(4), 1540009, 17 (2015)
Loi, A., Mossa, R.: Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds, 9 May 2023, preprint ar**v:2305.05244 [math.DG]
Loi, A., Placini, G., Zedda, M.: Immersions into Sasakian space forms, 9 May 2023, preprint ar**v:2305.05509 [math.DG]
Placini, G.: Sasakian immersions of Sasaki-Ricci solitons into Sasakian space forms. J. Geom. Phys. 166, 104265, 7 (2021)
Loi, A., Mossa, R.: Kähler immersions of Kähler-Ricci solitons into definite or indefinite complex space forms. Proc. Am. Math. Soc. 149(11), 4931–4941 (2021)
Loi, A., Mossa, R.: Holomorphic isometries into homogeneous bounded domains. Proc. Am. Math. Soc. 151(9), 3975–3984 (2023)
Loi, A., Mossa, R.: On holomorphic isometries into blow-ups of \({\mathbb{C} }^n\). Mediterr. J. Math. 20(4), 230 (2023)
Hamilton, R., Ricci, S.: The, flow on surfaces, Mathematics and general relativity. Contemp. Math., Santa Cruz, CA, vol. 71, Amer. Math. Soc. Providence, RI 1988, 237–262 (1986)
Loi, A., Zedda, M.: On Calabi’s diastasis function of the Cigar metric. J. Geom. Phys. 110, 269–276 (2016)
Charles, P.: Boyer and Krzysztof Galicki. Oxford Mathematical Monographs, Oxford University Press, Oxford, Sasakian Geometry (2008)
Boothby, W.M., Wang, H.C.: On contact manifolds. Ann. Math. 68, 721–734 (1958)
Bruce, L.: Reinhart, Foliaated manifolds with bundle-like metrics. Ann. Math. 69, 119–132 (1959)
Cao, H.-D.: Existence of gradient Kähler-Ricci solitons, Elliptic and parabolic methods in geometry. A K Peters. Wellesley, MA 1996, 1–16 (1994)
Cao, H.-D.: Limits of solutions to the Kähler-Ricci flow. J. Differ. Geom. 45(2), 257–272 (1997)
Huang, X., Yuan, Y.: Submanifolds of Hermitian symmetric spaces. Analysis and Geometry, Springer Proc. Math. Stat., vol. 127, Springer, Cham, 2015, pp. 197–206
Kotschwar, B.L.: A local version of Bando’s theorem on the real-analyticity of solutions to the Ricci flow. Bull. Lond. Math. Soc. 45(1), 153–158 (2013)
Calabi, E.: Isometric imbedding of complex manifolds. Ann. Math. 58, 1–23 (1953)
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RM and GP contributed equally in all parts of the process.
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The authors are supported by INdAM and GNSAGA—Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni and by GOACT–Funded by Fondazione di Sardegna. The second author is funded by the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.5–Call for tender No.3277 published on December 30, 2021 by the Italian Ministry of University and Research (MUR) funded by the European Union - NextGenerationEU. Project Code ECS0000038–Project Title eINS Ecosystem of Innovation for Next Generation Sardinia–CUP F53C22000430001- Grant Assignment Decree No. 1056 adopted on June 23, 2022 by (MUR).
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Mossa, R., Placini, G. Immersions of Sasaki–Ricci solitons into homogeneous Sasakian manifolds. Ann Glob Anal Geom 65, 8 (2024). https://doi.org/10.1007/s10455-023-09939-4
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DOI: https://doi.org/10.1007/s10455-023-09939-4
Keywords
- Sasaki–Ricci soliton
- Sasakian immersion
- Homogeneous Sasakian manifold
- \(\eta \)-Einstein manifold
- Kähler–Ricci soliton