Research ArticlesNo Access

On rigidity and scalar curvature of Einstein-type manifolds

Department of Mathematics, Payame Noor University, P. O. Box 19395-3697, Tehran, Iran

E-mail Address: ah.mirshafeazadeh@phd.pnu.ac.ir

and

Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave. 15914, Tehran, Iran

E-mail Address: bidabad@aut.ac.ir

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0219887818500731Cited by:0 (Source: Crossref)

Abstract

Here, the scalar curvature of an Einstein-type manifold or equivalently an almost Yamabe quasi-soliton is explicitly determined in terms of its soliton function and some rigidity theorems are obtained. Among the others its shown; if the soliton function is negative, then every compact conformal Yamabe quasi-soliton is isometric to the standard Euclidean sphere.

Keywords:

AMSC: 53C21, 53C25

References

1. B. Andrews and C. Hopper, The Ricci Flow in Riemannian Geometry (Springer-Verlag, Berlin, 2011). Crossref, Google Scholar
2. A. Barros and J. N. Gomes, A compact gradient generalized quasi-Einstein metric with constant scalar curvature, J. Math. Anal. Appl. 401 (2013) 702–705. Crossref, Web of Science, Google Scholar
3. B. Bidabad and M. Yar Ahmadi, On complete Yamabe solitons, Advances in Geometry. https://doi.org/10.1515/advgeom-2017-0018 (2017). Web of Science, Google Scholar
4. B. Bidabad and M. Yar Ahmadi, On compact Ricci solitons in Finsler geometry, Comptes Rendus Mathematique 353 (2015) 1023–1027. Crossref, Web of Science, Google Scholar
5. J. Bourguignon and J. Ezin, Scalar curvature functions in a conformal class of metrics and conformal transformations, Trans. Amer. Math. Soc. 301 (1987) 723–736. Crossref, Web of Science, Google Scholar
6. G. Catino, C. Mantegazza and L. Mazzieri, On the global structure of conformal gradient solitons with nonnegative Ricci curvature, Commun. Contemp. Math. 14(6) (2012) 1250045. Link, Web of Science, Google Scholar
7. G. Catino, P. Mastrolia, D. D. Monticelli and M. Rigoli, On the geometry of gradient Einstein-type manifolds, Pacific J. Math. 286(1) (2017) 39–67. Crossref, Web of Science, Google Scholar
8. P. Petersen and W. Wylie, Rigidity of gradient Ricci solitons, Pacific J. Math. 241(2) (2009) 329–345. Crossref, Web of Science, Google Scholar
9. S. Tanno and W. C. Weber, Closed conformal vector fields, J. Differential Geom. 3(3–4) (1969) 361–366. Crossref, Google Scholar
10. Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965) 251–275. Crossref, Web of Science, Google Scholar
11. L. F. Wang, On noncompact quasi Yamabe gradient solitons, Differential Geom. Appl. 31 (2013) 337–348. Crossref, Web of Science, Google Scholar
12. F. Warner, Foundations of Differentiable Manifolds and Lie Groups (Springer-Verlag, New York, 1970). Google Scholar