Back to Almost Ricci Solitons

Year 2023, , 208 - 214, 30.04.2023

Vladimir Rovenski , Sergey Stepanov , Irina Tsyganok

https://doi.org/10.36890/iejg.1223973

Abstract

In the paper, we study complete almost Ricci solitons using the concepts and methods of geometric dynamics and geometric analysis. In particular, we characterize Einstein manifolds in the class of complete almost Ricci solitons.
Then, we examine compact almost Ricci solitons using the orthogonal expansion of the Ricci tensor, this allows us to substantiate the concept of almost Ricci solitons.

Keywords

Almost Ricci soliton, energy density, infinitesimal harmonic transformation, conformal Killing vector

References

• [1] Arnold V.I., Klesin B.A.: Topological methods in hydrodynamics, Springer-Verlag, New York (1998).
• [2] Barros A., Gomes J.N., Rebeiro E.: A note on rigidity of almost Ricci soliton. Archiv der Mathematik, 100, 481–490 (2013).
• [3] Barros A., Batista R., Ribeiro jr. E.: Compact almost Ricci solitons with constant scalar curvature are gradient. Monatshefte für Mathematik, 174, 29–39 (2014).
• [4] Berard P.H.: From vanishing theorems to estimating theorems: the Bochner technique revisited. Bulletin of the American Mathematical Society, 19 (2), 371–406 (1988).
• [5] Besse A.L.: Einstein manifolds, Springer-Verlag, Berlin and Heidelberg (2008).
• [6] Caminha, A.: The geometry of closed conformal Killing vector fields on Riemannian spaces. Bull. Braz. Math. Soc. (N.S.), 41 (2), 277–300 (2011).
• [7] Chow B., Lu P., Ni L.: Hamilton’s Ricci flow, in Grad. Stud. in Math., 77, AMS, Providence, RI (2006).
• [8] Deshmukh S.: Almost Ricci solitons isometric to spheres. Int. J. of Geom. Methods in Modern Physics. 16 (5) 1950073 (2019).
• [9] Duggal K.L., Almost Ricci Solitons and Physical Applications. Int. Electronic J. of Geometry, 10 (2), 1–10 (2017).
• [10] Kar D., Majhi P.: Beta-almost Ricci solitons on almost co-Kähler manifolds. Korean J. Math., 27 (3), 691–705 (2019).
• [11] Kobayashi S., Nomizu K.: Foundations of differential geometry. Vol. I. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York (1993).
• [12] Morgan J., Tian G.: Ricci flow and Poincare conjecture, Clay Mathematics Monographs, 3. AMS, Providence, RI; Clay Mathematics Institute, Cambridge, MA (2007).
• [13] O’Neil B.: Semi-Riemannian geometry with applications to relativity, Academic Press, London (1983).
• [14] Patra D.S., Rovenski V.: Almost η-Ricci solitons on Kenmotsu manifolds. European J. Math., 7, 1753–1766 (2021).
• [15] Pigola S., Rigoli M., Rimoldi M., Setti A. G., Ricci almost solitons. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (4), 757–799 (2011).
• [16] Stepanov S.E., Shandra I.G.: Geometry of infinitesimal harmonic transformations. Annals of Global Analysis and Geometry, 24, 291–299 (2003).
• [17] Stepanov S. E., Shelepova V. N.: A note on Ricci solitons. Mathematical Notes, 86 : 3, 447–450 (2009).
• [18] Stepanov S.E., Tsyganok I.I., Mikeš J.: From infinitesimal harmonic transformations to Ricci solitons. Mathematica Bohemica, 138 (1), 25–36 (2013).
• [19] Udriste C.: Geometric dynamics, Kluwer Academic Publishers, Dordrecht (2002).
• [20] Yano K.: Integral formulas in Riemannian geometry, Marcel Dekker, New York (1970).
• [21] Yau, S.-T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J., 25, 659–670 (1976).