Ricci-Bourguignon solitons on real hypersurfaces in the complex projective space

Authors

  • Imsoon Jeong Department of Mathematics Education, Cheongju University, Chungcheongbuk-do 28503, Republic of Korea
  • Young Jin Suh Department of Mathematics and RIRCM, Kyungpook National University, Daegu 41566, Republic of Korea

DOI:

https://doi.org/10.33044/revuma.3849

Abstract

We give a complete classification of Ricci–Bourguignon solitons on real hypersurfaces in the complex projective space $\mathbb{C}P^n=SU_{n+1}/S(U_1 \cdot U_n)$. Next, as an application, we give some non-existence properties for gradient Ricci–Bourguignon solitons on real hypersurfaces with isometric Reeb flow and contact real hypersurfaces in the complex projective space $\mathbb{C}P^n$.

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References

J. Berndt and Y. J. Suh, Real hypersurfaces in Hermitian symmetric spaces, Advances in Analysis and Geometry 5, De Gruyter, Berlin, 2022.  DOI  MR  Zbl

A. M. Blaga and H. M. Taştan, Some results on almost $η$-Ricci–Bourguignon solitons, J. Geom. Phys. 168 (2021), article no. 104316.  DOI  MR  Zbl

D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509, Springer, Berlin-New York, 1976.  MR  Zbl

J.-P. Bourguignon, Une stratification de l'espace des structures riemanniennes, Compositio Math. 30 (1975), 1–41.  MR  Zbl

J.-P. Bourguignon, Ricci curvature and Einstein metrics, in Global differential geometry and global analysis (Berlin, 1979), Lecture Notes in Math. 838, Springer, Berlin, 1981, pp. 42–63.  MR  Zbl

G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza, and L. Mazzieri, The Ricci–Bourguignon flow, Pacific J. Math. 287 no. 2 (2017), 337–370.  DOI  MR  Zbl

G. Catino and L. Mazzieri, Gradient Einstein solitons, Nonlinear Anal. 132 (2016), 66–94.  DOI  MR  Zbl

G. Catino, L. Mazzieri, and S. Mongodi, Rigidity of gradient Einstein shrinkers, Commun. Contemp. Math. 17 no. 6 (2015), article no. 1550046.  DOI  MR  Zbl

T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 no. 2 (1982), 481–499.  DOI  MR  Zbl

P. Cernea and D. Guan, Killing fields generated by multiple solutions to the Fischer–Marsden equation, Internat. J. Math. 26 no. 4 (2015), article no. 1540006.  DOI  MR  Zbl

S. K. Chaubey, U. C. De, and Y. J. Suh, Kenmotsu manifolds satisfying the Fischer–Marsden equation, J. Korean Math. Soc. 58 no. 3 (2021), 597–607.  DOI  MR  Zbl

S. K. Chaubey, Y. J. Suh, and U. C. De, Characterizations of the Lorentzian manifolds admitting a type of semi-symmetric metric connection, Anal. Math. Phys. 10 no. 4 (2020), article no. 61.  DOI  MR  Zbl

U. C. De, S. K. Chaubey, and Y. J. Suh, Gradient Yamabe and gradient $m$-quasi Einstein metrics on three-dimensional cosymplectic manifolds, Mediterr. J. Math. 18 no. 3 (2021), article no. 80.  DOI  MR  Zbl

M. Djorić and M. Okumura, CR submanifolds of complex projective space, Developments in Mathematics 19, Springer, New York, 2010.  DOI  MR  Zbl

S. Dwivedi, Some results on Ricci–Bourguignon solitons and almost solitons, Canad. Math. Bull. 64 no. 3 (2021), 591–604.  DOI  MR  Zbl

R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math. 71, American Mathematical Society, Providence, RI, 1988, pp. 237–262.  DOI  MR  Zbl

S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics 34, American Mathematical Society, Providence, RI, 2001.  DOI  MR  Zbl

I. Jeong and Y. J. Suh, Pseudo anti-commuting and Ricci soliton real hypersurfaces in complex two-plane Grassmannians, J. Geom. Phys. 86 (2014), 258–272.  DOI  MR  Zbl

U.-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama Univ. 32 (1990), 207–221.  MR  Zbl

S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II, Wiley Classics Library, John Wiley & Sons, New York, 1996.  MR  Zbl

J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007.  MR  Zbl

M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975), 355–364.  DOI  MR  Zbl

B. O'Neill, Semi-Riemannian geometry, Pure and Applied Mathematics 103, Academic Press, New York, 1983.  MR  Zbl

G. Perelman, Ricci flow with surgery on three-manifolds, 2003. arXiv:math/0303109 [math.DG].

J. D. Pérez, Commutativity of Cho and structure Jacobi operators of a real hypersurface in a complex projective space, Ann. Mat. Pura Appl. (4) 194 no. 6 (2015), 1781–1794.  DOI  MR  Zbl

A. Romero, On a certain class of complex Einstein hypersurfaces in indefinite complex space forms, Math. Z. 192 no. 4 (1986), 627–635.  DOI  MR  Zbl

A. Romero, Some examples of indefinite complete complex Einstein hypersurfaces not locally symmetric, Proc. Amer. Math. Soc. 98 no. 2 (1986), 283–286.  DOI  MR  Zbl

B. Smyth, Differential geometry of complex hypersurfaces, Ann. of Math. (2) 85 (1967), 246–266.  DOI  MR  Zbl

Y. J. Suh, Real hypersurfaces of type $B$ in complex two-plane Grassmannians, Monatsh. Math. 147 no. 4 (2006), 337–355.  DOI  MR  Zbl

Y. J. Suh, Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians, Adv. in Appl. Math. 50 no. 4 (2013), 645–659.  DOI  MR  Zbl

Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with harmonic curvature, J. Math. Pures Appl. (9) 100 no. 1 (2013), 16–33.  DOI  MR  Zbl

R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495–506.  MR  Zbl Available at http://projecteuclid.org/euclid.ojm/1200694557.

Y. Wang, Ricci solitons on almost Kenmotsu 3-manifolds, Open Math. 15 no. 1 (2017), 1236–1243.  DOI  MR  Zbl

Y. Wang, Ricci solitons on almost co-Kähler manifolds, Canad. Math. Bull. 62 no. 4 (2019), 912–922.  DOI  MR  Zbl

K. Yano and M. Kon, CR submanifolds of Kaehlerian and Sasakian manifolds, Progress in Mathematics 30, Birkhäuser, Boston, MA, 1983.  MR  Zbl

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Published

2025-05-21

Issue

Vol. 68 No. 1 (2025)

Section

Article

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