Characterizations and integral formulae for generalized m-quasi-Einstein metrics

  • Abdênago BarrosEmail author
  • Ernani RibeiroJr


The aim of this paper is to present some structural equations for generalized m-quasi-Einstein metrics (M n , g, ∇ f, λ), which was defined recently by Catino in [11]. In addition, supposing that M n is an Einstein manifold we shall show that it is a space form with a well defined potential f. Finally, we shall derive a formula for the Laplacian of its scalar curvature which will give some integral formulae for such a class of compact manifolds that permit to obtain some rigidity results.


Ricci soliton quasi-Einstein metrics Bakry-Emery Ricci tensor scalar curvature 

Mathematical subject classification

Primary: 53C25, 53C20, 53C21 Secondary: 53C65 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Anderson. Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds, I. Geom. Funct. Anal., 9 (1999), 855–967.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    M. Anderson and M. Khuri. The static extension problem in General relativity. arXiv:0909.4550v1 [math.DG], (2009).Google Scholar
  3. [3]
    C. Aquino, A. Barros and E. Ribeiro Jr. Some applications of the Hodge-de Rham decomposition to Ricci solitons. Results inMath., 60 (2011), 235–246.CrossRefMathSciNetGoogle Scholar
  4. [4]
    A. Barros and E. Ribeiro Jr. Some characterizations for compact almost Ricci solitons. Proc. Amer.Math. Soc., 140 (2012), 1033–1040.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    A. Barros and E. Ribeiro Jr. Integral formulae on quasi-Einstein manifolds and applications. Glasgow Math. J., 54 (2012), 213–223.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    J.P. Bourguignon and J.P. Ezin. Scalar curvature functions in a conformal class of metrics and conformal transformations. Trans. Amer. Math. Soc., 301 (1987), 723–736.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    A. Caminha, F. Camargo and P. Souza. Complete foliations of space forms by hypersurfaces. Bull. Braz. Math. Soc., 41 (2010), 339–353.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    H.D. Cao. Recent progress on Ricci soliton. Adv. Lect. Math., 11 (2009), 1–38.Google Scholar
  9. [9]
    J. Case, Y. Shu and G. Wei. Rigity of quasi-Einstein metrics. Differ. Geom. Appl., 29 (2011), 93–100.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    J. Case. On the nonexistence of quasi-Einstein metrics. Pacific J. Math., 248 (2010), 227–284.CrossRefMathSciNetGoogle Scholar
  11. [11]
    G. Catino. Generalized quasi-Einstein manifolds with harmonic weyl tensor. Math. Z., 271 (2012), 751–756.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    J. Corvino. Scalar curvature deformations and a gluing construction for the Einstein constraint equations. Comm. Math. Phys., 214 (2000), 137–189.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    R.S. Hamilton. The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7–136, International Press, Cambridge, MA (1995).Google Scholar
  14. [14]
    C. He, P. Petersen and W. Wylie. On the classification of warped product Einstein metrics. Commun. in Analysis and Geometry, 20 (2012), 271–312.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    P. Petersen and W. Wylie. Rigidity of gradient Ricci solitons. Pacific J. Math., 241–2 (2009), 329–345.CrossRefMathSciNetGoogle Scholar
  16. [16]
    S. Pigola, M. Rigoli, M. Rimoldi and A. Setti. Ricci Almost Solitons. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. X (2011), 757–799.MathSciNetGoogle Scholar
  17. [17]
    Y. Tashiro. Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc., 117 (1965), 251–275.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    Yau S.T. Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J., 25 (1976), 659–670.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2014

Authors and Affiliations

  1. 1.Departamento de MatemáticaUFCFortaleza, CEBrazil

Personalised recommendations