Quasi Yamabe Solitons on CR Submanifolds of Maximal CR Dimension in Kähler Manifolds

Year 2024, Volume: 17 Issue: 1, 267 - 276, 23.04.2024

Milos B Djoric Mirjana Djoric

https://doi.org/10.36890/iejg.1454012

Abstract

In this paper we give necessary and sufficient conditions for a CR submanifold of maximal CR
dimension in arbitrary Kähler manifold to admit (quasi-)Yamabe structure, with naturally chosen
soliton vector field. When the ambient manifold is a non-flat complex space form, we give a
complete classification of such solitons, under certain conditions.

Keywords

Yamabe soliton, quasi Yamabe soliton, complex space form, CR submanifold of maximal CR dimension, structure vector field, constant scalar curvature

References

• [1] Bejancu, A.: CR-submanifolds of a Kähler manifold I. Proc. Amer. Math. Soc. 69, 135-142 (1978).
• [2] Chen, B.-Y., Deshmukh, S.: Yamabe and quasi-Yamabe solitons on Euclidean submanifolds. Mediterr. J. Math. 15 (194), (2018).
• [3] Chen, B.-Y., Djoric, M. B., Djoric, M.: Quasi-Yamabe and Yamabe solitons on hypersurfaces of nearly Kähler manifolds. Mediterr. J. Math. 21 (10), (2024).
• [4] Cho, J. T., Kimura, M.: Ricci solitons and real hypersurfaces in a complex space form. Tohoku Math. J. 61 (2), 205-212 (2009).
• [5] Cho, J. T., Kimura, M.: Ricci solitons of compact real hypersurfaces in Kähler manifolds. Math. Nachr. 284 (11-12), 1385-1393 (2011).
• [6] Djoric, M., Okumura, M.: Certain CR submanifolds of maximal CR dimension of complex space forms. Differential Geom. Appl. 26, 208-217 (2008).
• [7] Djoric, M., Okumura, M.: Scalar curvature of CR submanifolds of maximal CR dimension of complex projective space. Monatsh. Math. 154, 11-17 (2008).
• [8] Djoric, M., Okumura, M.: CR submanifolds of complex projective space. Developments in Mathematics. 19 Springer, New York (2009).
• [9] Hamilton, R. S.: The Ricci flow on surfaces. Math. Gen. Relativ. (Santa Cruz, CA, 1986), Contemp. Math. 71 Amer. Math. Soc. 237-262 (1988).
• [10] Huang, G., Li, H.: On a classification of the quasi Yamabe gradient solitons. Methods Appl. Anal. 21 (3), 379-389 (2014).
• [11] Kawamoto, S.: Codimension reduction for real submanifolds of complex hyperbolic space. Revista Mathematica de la Universidad Complutense de Madrid 7, 119-128 (1994).
• [12] Leandro, B., Pina, H.: Generalized quasi Yamabe gradient solitons. J. Diff. Geom. Appl. 49, 167-175 (2016).
• [13] Lee, J. M., Parker, T. H.: The Yamabe problem, Bull of Amer. Math. Soc. 17 (1), 37-91 (1987).
• [14] Montiel, S., Romero, A.: On some real hypersurfaces of a complex hyperbolic space. Geom. Dedicata 20, 245-261 (1986).
• [15] Niebergall, R., Ryan, P. J.: Real hypersurfaces in complex space forms. In: Tight and taut submanifolds. (eds. T. E. Cecil and S.-S. Chern) Math. Sci. Res. Inst. Publ. 32 Cambridge University Press, Cambridge 233-305 (1997).
• [16] Okumura, M.: On some real hypersurfaces of a complex projective space. Trans. Amer. Math. Soc. 212, 355-364 (1975).
• [17] Okumura, M.: Codimension reduction problem for real submanifolds of complex projective space. Colloq. Math. Soc. János Bolyai 56, 574-585 (1989).