Harmonic Aspects in an $\eta$-Ricci Soliton

Year 2020, Volume: 13 Issue: 1, 41 - 49, 30.01.2020

Adara-monica Blaga

https://doi.org/10.36890/iejg.573919

Cited By: 5

Abstract

We characterize the $\eta$-Ricci solitons $(g,\xi,\lambda,\mu)$ for the special cases when the $1$-form $\eta$, which is the $g$-dual of $\xi$, is a harmonic or a Schr\"{o}dinger-Ricci harmonic form. We also provide necessary and sufficient conditions for $\eta$ to be a solution of the Schr\"{o}dinger-Ricci equation and point out the relation between the three notions in our context. In particular, we apply these results to a perfect fluid spacetime and using Bochner-Weitzenb\"{o}ck techniques, we formulate some more conclusions for the case of gradient solitons and deduce topological properties of the manifold and its universal covering.

Keywords

gradient Ricci solitons, Schrödinger-Ricci equation, harmonic form

References

• [1] Blaga, A. M.: A note on almost η-Ricci solitons in Euclidean hypersurfaces. Serdica Math. J. 43 (3-4), 361-368 (2017).
• [2] Blaga, A. M.: A note on warped product almost quasi-Yamabe solitons. Filomat 33 (7), 2009-2016 (2019).
• [3] Blaga, A. M.: Almost η-Ricci solitons in (LCS)n-manifolds. Bull. Belg. Math. Soc. Simon Stevin 25 (5), 641-653 (2018).
• [4] Blaga, A. M.: η-Ricci solitons on Lorentzian para-Sasakian manifolds. Filomat 30 (2), 489-496 (2016).
• [5] Blaga, A. M.: η-Ricci solitons on para-Kenmotsu manifolds. Balkan J. Geom. Appl. 20 (1), 1-13 (2015).
• [6] Blaga, A. M.: Last multipliers on η-Ricci solitons. Matematichki Bilten 42 (2), 85-90 (2018).
• [7] Blaga, A. M.: On gradient η-Einstein solitons. Kragujevak J. Math. 42 (2), 229-237 (2018).
• [8] Blaga, A. M.: On harmonicity and Miao-Tam critical metrics in a perfect fluid spacetime. To appear in Bol. Soc. Mat. Mexicana.
• [9] Blaga, A. M.: On solitons in statistical geometry. Int. J. Appl. Math. Stat. 58 (4), (2019).
• [10] Blaga, A. M.: On warped product gradient η-Ricci solitons. Filomat 31 (18), 5791-5801 (2017).
• [11] Blaga, A. M.: Remarks on almost η-solitons. Matematicki Vesnik 71 (3), 244-249 (2019).
• [12] Blaga, A. M.: Solitons and geometrical structures in a perfect fluid spacetime. To appear in Rocky Mountain J. Math.
• [13] Blaga, A. M.: Solutions of some types of soliton equations in R3. Filomat 33 (4), 1159-1162 (2019).
• [14] Blaga, A. M.: Some geometrical aspects of Einstein, Ricci and Yamabe solitons. J. Geom. Sym. Phys. 52, 17-26 (2019).
• [15] Blaga, A. M., Baishya, K. K., Sarkar, N.: Ricci solitons in a generalized weakly (Ricci) symmetric D-homothetically deformed Kenmotsu manifold. Ann. Univ. Paedagog. Crac. Stud. Math. 18, 123-136 (2019).
• [16] Blaga, A. M., Crasmareanu, M. C.: Inequalities for gradient Einstein and Ricci solitons. To appear in Facta Univ. Math. Inform.
• [17] Blaga, A. M., Crasmareanu, M. C.: Torse-forming η-Ricci solitons in almost paracontact η-Einstein geometry. Filomat 31 (2), 499-504 (2017).
• [18] Blaga, A. M., Perktaş, S. Y.: Remarks on almost η-Ricci solitons in ε-para Sasakian manifolds. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68 (2), 1621-1628 (2019).
• [19] Blaga, A. M., Perktaş, S. Y., Acet, B. E., Erdogan, F. E.: η-Ricci solitons in ε-almost paracontact metric manifolds. Glasnik Matematicki 53 (1), 377-410 (2018).
• [20] Catino, G.: A note on four-dimensional (anti-)self-dual quasi-Einstein manifolds. Differential Geom. Appl. 30, 660-664 (2012).
• [21] Călin, C., Crasmareanu, M.: Eta-Ricci solitons on Hopf hypersurfaces in complex space forms. Rev. Roumaine Math. Pures Appl. 57 (1), 55-63 (2012).
• [22] Cho, J. T., Kimura, M.: Ricci solitons and real hypersurfaces in a complex space form. Tohoku Math. J. 61 (2), 205-212 (2009).
• [23] Chaki, M. C., Maity, R. K.: On quasi Einstein manifolds. Publ. Math. Debrecen 57, 297-306 (2000).
• [24] Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci Flow: Techniques and Applications: Part I: Geometric Aspects 135. AMS (2007).
• [25] Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. Graduate Studies in Mathematics 77. AMS (2006).
• [26] De, K., Blaga, A. M., De, U. C.: ∗-Ricci solitons on (ε)-Kenmotsu manifolds. To appear in Palestine Math. J.
• [27] De, U. C., Ghosh, G. C.: On quasi-Einstein and special quasi-Einstein manifolds, Proc. of the Int. Conf. of Mathematics and its Applications, Kuwait University, April 5-7, 2004, 178-191.
• [28] De, U. C., Ghosh, G. C.: On quasi-Einstein manifolds. Period. Math. Hungar. 48 (1-2), 223-231 (2004).
• [29] Deng, H.: Compact manifolds with positive m-Bakry-Émery Ricci tensor. Differential Geom. Appl. 32, 88-96 (2014).
• [30] Deszcz, R., Hotlos, M., Senturk, Z.: On curvature properties of quasi-Einstein hypersurfaces in semi-Euclidean spaces. Soochow J. Math. 27, 375-389 (2001).
• [31] Hamilton, R. S.: The Ricci flow on surfaces, in Mathematics and General Relativity, Contemp. Math. 71. AMS, 237-262 (1988).
• [32] Hamilton, R. S.: Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (2), 255-306 (1982).
• [33] Lima, E. L.: Grupo Fundamental e Espaços de Recobrimento. IMPA (2012).
• [34] Lot, J.: Some geometric properties of the Bakry-Émery Ricci tensor. Comment. Math. Helv. 78 (4), 865-883 (2003).
• [35] Petersen, P., Wylie, W.: Rigidity of gradient Ricci solitons. Pacific J. Math. 241 (2), 329-345 (2009).
• [36] Qian, Z.: Estimates for weighted volumes and applications. Quart. J. Math. Oxford Ser. (2) 48 (190), 235-242 (1997).
• [37] Smale, S.: On gradient dynamical systems. Ann. of Math. 2 (74), 199-206 (1961).
• [38] Vieira, M.: Harmonic forms on manifolds with non-negative Bakry-Émery Ricci curvature. Archiv der Mathematik 101 (6), 581-590 (2013).