A potential generalization of some canonical Riemannian metrics

  • Giovanni Catino
  • Paolo MastroliaEmail author


The aim of this paper is to study new classes of Riemannian manifolds endowed with a smooth potential function, including in a general framework classical canonical structures such as Einstein, harmonic curvature and Yamabe metrics, and, above all, gradient Ricci solitons. For the most rigid cases, we give a complete classification, while for the others we provide rigidity and obstruction results, characterizations and nontrivial examples. In the final part of the paper, we also describe the “nongradient” version of this construction.


Canonical metrics Einstein metrics Harmonic curvature Yamabe metrics Ricci solitons 

Mathematics Subject Classification

53C20 53C25 



The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and they are partially supported by GNAMPA project “Strutture di tipo Einstein e Analisi Geometrica su varietà Riemanniane e Lorentziane.”


  1. 1.
    Alias, L.J., Mastrolia, P., Rigoli, M.: Maximum Principles and Geometric Applications. Springer Monographs in Mathematics. Springer, Berlin (2016)CrossRefzbMATHGoogle Scholar
  2. 2.
    Aubin, T.: Métriques riemanniennes et courbure. J. Differential Geom. 4, 383–424 (1970)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bach, R.: Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krummungstensorbegriffs. Math. Z. 9, 110–135 (1921)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baird, P.: A class of three-dimensional Ricci solitons. Geom. Topol. 13, 979–1015 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baird, P., Danielo, L.: Three-dimensional Ricci solitons which project to surfaces. J. Reine Angew. Math. 608, 65–91 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Besse, A.L.: Einstein Manifolds. Springer, Berlin (2008)zbMATHGoogle Scholar
  7. 7.
    Bland, J., Kalka, M.: Negative scalar curvature metrics on noncompact manifolds. Trans. Amer. Math. Soc. 316(2), 433–446 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Böhm, C., Wilking, B.: Manifolds with positive curvature operators are space forms. Ann. Math. (2) 167(3), 1079–1097 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bourguignon, J.-B., Ezin, J.-P.: Scalar curvature functions in a conformal class of metrics and conformal transformations. Trans. Amer. Math. Soc. 301(2), 723–736 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bourguignon, J.-P.: Les variétés de dimension \(4\) à signature non nulle dont la courbure est harmonique sont d’Einstein. Invent. Math. 63(2), 263–286 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bourguignon, J.P.: Ricci curvature and Einstein metrics. In: Ferus, D., Kühnel, W., Simon, U., Wegner, B. (eds.) Global Differential Geometry and Global Analysis. Lecture Notes in Mathematics, vol. 838, pp. 42–63. Springer, Berlin (1981)CrossRefGoogle Scholar
  12. 12.
    Bourguignon, J.-P., Jr, H.B.L.: Stability and isolation phenomena for Yang–Mills fields. Comm. Math. Phys. 79(2), 189–230 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bryant, R.L.: Ricci flow solitons in dimension three with \({S}{O}(3)\)–symmetries. (2005)
  14. 14.
    Bueler, E.L.: The heat kernel weighted Hodge Laplacian on noncompact manifolds. Trans. Amer. Math. Soc. 351(2), 683–713 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cao, H.-D.: Recent progress on Ricci solitons. Recent Advances in Geometric Analysis. Advanced Lectures in Mathematics, vol. 11, pp. 1–38. International Press, Somerville (2010)Google Scholar
  16. 16.
    Cao, H.-D., Chen, Q.: On locally conformally flat gradient steady Ricci solitons. Trans. Amer. Math. Soc. 364(5), 2377–2391 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cao, H.-D., Chen, Q.: On Bach-flat gradient shrinking Ricci solitons. Duke Math. J. 162(6), 1149–1169 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cartan, E.: Sur une classe remarquable d’espaces de Riemann. II. Bull. Soc. Math. France 55, 114–134 (1927)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Cartan, E.: Leçons sur la géométrie des espaces de Riemann, 2nd edn. Gauthier-Villars, Paris (1951)zbMATHGoogle Scholar
  20. 20.
    Catino, G.: On conformally flat manifolds with constant positive scalar curvature. Proc. Amer. Math. Soc. 144(6), 2627–2634 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Catino, G., Mantegazza, C.: The evolution of the Weyl tensor under the Ricci flow. Ann. Inst. Fourier (Grenoble) 61(4), 1407–1435 (2012). 2011MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Catino, G., Mantegazza, C., Mazzieri, L.: Locally conformally flat ancient Ricci flows. Anal. PDE 8(2), 365–371 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Catino, G., Mantegazza, C., Mazzieri, L.: A note on Codazzi tensors. Math. Ann. 362(1–2), 629–638 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Catino, G., Mastrolia, P., Monticelli, D.D.: Gradient Ricci solitons with vanishing conditions on Weyl. J. Math. Pures Appl. (9) 108(1), 1–13 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Catino, G., Mastrolia, P., Monticelli, D.D., Rigoli, M.: Conformal Ricci solitons and related integrability conditions. Adv. Geom. 16(3), 301–328 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Chen, X., Wang, Y.: On four-dimensional anti-self-dual gradient Ricci solitons. J. Geom. Anal. 25(2), 1335–1343 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Derdziński, A.: On compact Riemannian manifolds with harmonic curvature. Math. Ann. 259(2), 145–152 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Drdziński, A.: Riemannian manifolds with harmonic curvature. In: Ferus, D., Gardner, R.B., Helgason, S., Simon, U. (eds.) Global Differential Geometry and Global Analysis 1984. Lecture Notes in Mathematics, vol. 1156, pp. 74–85. Springer, Berlin (1985)CrossRefGoogle Scholar
  29. 29.
    Derdziński, A., Roter, W.: On conformally symmetric manifolds with metrics of indices \(0\) and \(1\). Tensor (N.S.) 31(3), 255–259 (1977)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Derdziński, A., Shen, C.L.: Codazzi tensor fields, curvature and Pontryagin forms. Proc. Lond. Math. Soc. (3) 47(1), 15–26 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    DeTurck, D., Goldschmidt, H.: Regularity theorems in Riemannian geometry. II. Harmonic curvature and the Weyl tensor. Forum Math. 1(4), 377–394 (1989)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Eminenti, M., La Nave, G., Mantegazza, C.: Ricci solitons: the equation point of view. Manuscripta Math. 127(3), 345–367 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Fernández-López, M., García-Río, E.: Rigidity of shrinking Ricci solitons. Math. Z. 269(1–2), 461–466 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry. Springer, Berlin (1990)CrossRefzbMATHGoogle Scholar
  35. 35.
    Gursky, M.J.: Conformal vector fields on four-manifolds with negative scalar curvature. Math. Z. 232(2), 265–273 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differential Geom. 17(2), 255–306 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kazdan, J.L.: Unique continuation in geometry. Comm. Pure Appl. Math. 41, 667–681 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kazdan, J.L., Warner, F.W.: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures. Ann. Math. 2(101), 317–331 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Kazdan, J.L., Warner, F.W.: Scalar curvature and conformal deformation of Riemannian structure. J. Differential Geom. 10, 113–134 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Lauret, J.: Ricci soliton homogeneous nilmanifolds. Math. Ann. 319, 715–733 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Amer. Math. Soc. (N.S.) 17(1), 37–91 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Lott, J.: On the long-time behavior of type-III Ricci flow solutions. Math. Ann. 339(3), 627–666 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Merton, G.: Codazzi tensors with two eigenvalue functions. Proc. Amer. Math. Soc. 141(9), 3265–3273 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Naber, A.: Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math. 645, 125–153 (2010)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Ni, L., Wallach, N.: On a classification of gradient shrinking solitons. Math. Res. Lett. 15(5), 941–955 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Ozeki, K.N.H.: A theorem on curvature tensor fields. Proc. Nat. Acad. Sci. USA 48, 206–207 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. (2002)
  48. 48.
    Petersen, P., Wylie, W.: On the classification of gradient Ricci solitons. Geom. Topol. 14, 2277–2300 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Tachibana, S.: A theorem of Riemannian manifolds of positive curvature operator. Proc. Japan Acad. 50, 301–302 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Tanno, S.: Curvature tensors and covariant derivatives. Ann. Mat. Pura Appl. 4(96), 233–241 (1972)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc 117, 251–275 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Yano, K., Bochner, S.: Curvature and Betti Numbers. Annals of Mathematics Studies, vol. 32. Princeton University Press, Princeton (1953)zbMATHGoogle Scholar
  53. 53.
    Zhang, Z.-H.: Gradient shrinking solitons with vanishing Weyl tensor. Pacific J. Math. 242(1), 189–200 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  2. 2.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanItaly

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