On the rigidity of harmonic-Ricci solitons

References

[1] A. Abolarinwa, Basic structural equations for almost Ricci-harmonic solitons and applications. Differ. Geom. Dyn. Syst. 21 (2019), 1–13. MR3997841 Zbl 1423.53055Search in Google Scholar

[2] L. J. Alías, P. Mastrolia, M. Rigoli, Maximum principles and geometric applications. Springer 2016. MR3445380 Zbl 1346.5800110.1007/978-3-319-24337-5Search in Google Scholar

[3] A. Anselli, Phi-curvatures, harmonic-Einstein manifolds and Einstein-type structures. PhD thesis, Universitá degli Studi di Milano, 2020.Search in Google Scholar

[4] A. Anselli, Bach and Einstein’s equations in presence of a field. Int. J. Geom. Methods. Mod. Phys. 18 (2021), no. 5, 215007. MR425475910.1142/S0219887821500778Search in Google Scholar

[5] A. Anselli, G. Colombo, M. Rigoli, On the geometry of Einstein-type structures. Nonlinear Anal. 204 (2021), 112198. MR4184679 Zbl 731096310.1016/j.na.2020.112198Search in Google Scholar

[6] C. Aquino, A. Barros, E. Ribeiro, Jr., Some applications of the Hodge–de Rham decomposition to Ricci solitons. Results Math. 60 (2011), 245–254. MR2836897 Zbl 1252.5305410.1007/s00025-011-0166-1Search in Google Scholar

[7] R. Bach, Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs. Math. Z. 9 (1921), 110–135. MR1544454 JFM 48.1035.0110.1007/BF01378338Search in Google Scholar

[8] P. Baird, J. Eells, A conservation law for harmonic maps. In: Geometry Symposium, Utrecht 1980 (Utrecht, 1980), volume 894 of Lecture Notes in Math., 1–25, Springer 1981. MR655417 Zbl 0485.5800810.1007/BFb0096222Search in Google Scholar

[9] P. Baird, J. C. Wood, Harmonic morphisms between Riemannian manifolds, volume 29 of London Mathematical Society Monographs. New Series. Oxford Univ. Press 2003. MR2044031 Zbl 1055.5304910.1093/acprof:oso/9780198503620.001.0001Search in Google Scholar

[10] A. Barros, R. Batista, E. Ribeiro, Jr., Compact almost Ricci solitons with constant scalar curvature are gradient. Monatsh. Math. 174 (2014), 29–39. MR3190769 Zbl 1296.5309210.1007/s00605-013-0581-3Search in Google Scholar

[11] E. Batista, L. Adriano, W. Tokura, Gradient Einstein-type structures immersed into a Riemannian warped product. To appear in J. Geom. Phys., 104510.10.1016/j.geomphys.2022.104510Search in Google Scholar

[12] F. Brito, H. L. Liu, U. Simon, C. P. Wang, Hypersurfaces in space forms with some constant curvature functions. In: Geometry and topology of submanifolds, IX (Valenciennes/Lyon/Leuven, 1997), 48–63, World Scientific, Singapore 1999. MR1787369 Zbl 0926.0002810.1142/9789812817976_0006Search in Google Scholar

[13] H.-D. Cao, Recent progress on Ricci solitons. In: Recent advances in geometric analysis, volume 11 of Adv. Lect. Math., 1–38, Int. Press, Somerville, MA 2010. MR2648937 Zbl 1201.53046Search in Google Scholar

[14] H.-D. Cao, Q. Chen, On Bach-flat gradient shrinking Ricci solitons. Duke Math. J. 162 (2013), 1149–1169. MR3053567 Zbl 1277.5303610.1215/00127094-2147649Search in Google Scholar

[15] G. de Rham, Sur la reductibilité d’un espace de Riemann. Comment. Math. Helv. 26 (1952), 328–344. MR52177 Zbl 0048.1570110.1007/BF02564308Search in Google Scholar

[16] G. Di Matteo, Analysis of Type I Singularities in the Harmonic Ricci Flow. Preprint 2018, arXiv:1811.09563 [math.DG]Search in Google Scholar

[17] M. Eminenti, G. La Nave, C. Mantegazza, Ricci solitons: the equation point of view. Manuscripta Math. 127 (2008), 345–367. MR2448435 Zbl 1160.5303110.1007/s00229-008-0210-ySearch in Google Scholar

[18] M. Fernández-López, E. García-Río, Rigidity of shrinking Ricci solitons. Math. Z. 269 (2011), 461–466. MR2836079 Zbl 1226.5304710.1007/s00209-010-0745-ySearch in Google Scholar

[19] R. S. Hamilton, Three-manifolds with positive Ricci curvature. J. Differential Geometry17 (1982), 255–306. MR664497 Zbl 0504.5303410.4310/jdg/1214436922Search in Google Scholar

[20] D. D. Joyce, Riemannian holonomy groups and calibrated geometry. Oxford Univ. Press 2007. MR2292510 Zbl 1200.5300310.1093/oso/9780199215607.001.0001Search in Google Scholar

[21] B. List, Evolution of an extended Ricci flow system. Comm. Anal. Geom. 16 (2008), 1007–1048. MR2471366 Zbl 1166.5304410.4310/CAG.2008.v16.n5.a5Search in Google Scholar

[22] R. Müller, Ricci flow coupled with harmonic map flow. Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), 101–142. MR2961788 Zbl 1247.5308210.24033/asens.2161Search in Google Scholar

[23] O. Munteanu, N. Sesum, On gradient Ricci solitons. J. Geom. Anal. 23 (2013), 539–561. MR3023848 Zbl 1275.5306110.1007/s12220-011-9252-6Search in Google Scholar

[24] A. Naber, Some geometry and analysis on Ricci solitons. Preprint 2006, arXiv:math/0612532 [math.DG]Search in Google Scholar

[25] P. Petersen, W. Wylie, Rigidity of gradient Ricci solitons. Pacific J. Math. 241 (2009), 329–345. MR2507581 Zbl 1176.5304810.2140/pjm.2009.241.329Search in Google Scholar

[26] P. Shi, Singularities of Connection Ricci Flow and Ricci Harmonic Flow. Preprint 2013, arXiv:1309.5684 [math.DG]Search in Google Scholar

[27] Y. Tashiro, Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117 (1965), 251–275. MR174022 Zbl 0136.1770110.1090/S0002-9947-1965-0174022-6Search in Google Scholar

[28] L. F. Wang, On Ricci-harmonic metrics. Ann. Acad. Sci. Fenn. Math. 41 (2016), 417–437. MR3467719 Zbl 1342.5305910.5186/aasfm.2016.4127Search in Google Scholar

[29] F. Yang, J. Shen, Volume growth for gradient shrinking solitons of Ricci-harmonic flow. Sci. China Math. 55 (2012), 1221–1228. MR2925588 Zbl 1270.5308710.1007/s11425-012-4361-7Search in Google Scholar

[30] F. Yang, L.-D. Zhang, On complete shrinking Ricci-harmonic solitons. J. of Math. 36 (2016), 494–500.Search in Google Scholar

[31] Z.-H. Zhang, On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc. 137 (2009), 2755–2759. MR2497489 Zbl 1176.5304610.1090/S0002-9939-09-09866-9Search in Google Scholar