On Gauss–Bonnet Curvatures

Mohammed Larbi Labbi

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

The $(2k)$-th Gauss–Bonnet curvature is a generalization to higher dimensions of the $(2k)$-dimensional Gauss–Bonnet integrand, it coincides with the usual scalar curvature for $k=1$. The Gauss–Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where they are known as the Lagrangian of Lovelock gravity, Gauss–Bonnet Gravity and Lanczos gravity. In this paper we present various aspects of these curvature invariants and review their variational properties. In particular, we discuss natural generalizations of the Yamabe problem, Einstein metrics and minimal submanifolds.

Détail

Keywords: Gauss–Bonnet curvatures; Gauss–Bonnet gravity; lovelock gravity; generalized Einstein metrics; generalized minimal submanifolds; generalized Yamabe problem.

Bibliographie

[1] Adler R. J., “On excursion sets, tube formulas and maxima of random fields”, Ann. Appl. Probab., 10 (2000), 1–74 | MR | Zbl

[2] Albin P., Renormalizing curvature integrals on Poincaré–Einstein manifolds, math.DG/0504161 | MR

[3] Berger M., “Quelques formules de variation pour une structure riemannienne”, Ann. Sci. Ecole Norm. Sup. (4), 3 (1970), 285–294 | MR | Zbl

[4] Bernig A., “Variations of curvatures of subanalytic spaces and Schläfli-type formulas”, Ann. Global Anal. Geom., 24 (2003), 67–93 | DOI | MR | Zbl

[5] Bernig A., On some aspects of curvature, available at http://homeweb1.unifr.ch/BernigA/pub/

[6] Besse A. L., Einstein manifolds, Springer-Verlag, Berlin, 1987 | MR

[7] Bourguignon J. P., “Les variétés de dimension $4$ à signature non nulle dont la courbure est harmonique sont d'Einstein”, Invent. Math., 63 (1981), 263–286 | DOI | MR | Zbl

[8] Cai R. G., Ohta N., “Black holes in pure lovelock gravities”, Phys. Rev. D, 74 (2006), 064001, 8 pp., ages ; hep-th/0604088 | DOI | MR

[9] Cheeger J., Müller W., Schräder R., “On the curvature of piecewise flat spaces”, Comm. Math. Phys., 92 (1984), 405–454 | DOI | MR | Zbl

[10] Deruelle N., Madore J., On the quasi-linearity of the Einstein–Gauss–Bonnet gravity field equations, gr-qc/0305004

[11] Fenchel W., “On total curvatures of Riemannian manifolds. I”, J. Lond. Math. Soc., 15 (1940), 15–22 | DOI | MR

[12] Gray A., “Some relations between curvature and characteristic classes”, Math. Ann., 184 (1970), 257–267 | DOI | MR | Zbl

[13] Gray A., Tubes, Progress in Mathematics, 221, Birkhäuser Verlag, Basel, 2004 | MR

[14] Ishihara T., “Kinematic formulas for Weyl's curvature invariants on submanifolds in complex projective space”, Proc. Amer. Math. Soc., 97 (1986), 483–487 | DOI | MR | Zbl

[15] Kowalski O., “On the Gauss–Kronecker curvature tensors”, Math. Ann., 203 (1973), 335–343 | DOI | MR | Zbl

[16] Kulkarni R. S., “On Bianchi identities”, Math. Ann., 199 (1972), 175–204 | DOI | MR | Zbl

[17] Labbi M. L., “Double forms, curvature structures and the $(p,q)$-curvatures”, Trans. Amer. Math. Soc., 357 (2005), 3971–3992 ; math.DG/0404081 | DOI | MR | Zbl

[18] Labbi M. L., “On compact manifolds with positive second Gauss–Bonnet curvature”, Pacific J. Math., 227 (2006), 295–310 | DOI | MR | Zbl

[19] Labbi M. L., “Variational properties of the Gauss–Bonnet curvatures”, Calculus of Variations and Partial Differential Equations, 32:2 (2008), 175–189 ; ; Labbi M. L., Riemannian curvature: variations on different notions of positivity, math.DG/0406548math.DG/0611371 | DOI | MR | Zbl

[20] Labbi M. L., On Weitzenböck curvature operators, math.DG/0607521

[21] Labbi M. L., Remarks on generalized Einstein manifolds, math.DG/0703028

[22] Labbi M. L., On $(2k)$-minimal submanifolds, arXiv:0706.3092 | MR

[23] Lafontaine J., “Mesures de courbure des variétés lisses et des polyèdres [d'après Cheeger, Müller et Schräder]”, Séminaire Bourbaki, 38ème année, 1985–86, Exp. No. 664, Asterisque, 145/146, 1987, 241–256 | MR | Zbl

[24] Lovelock D., “The Einstein tensor and its generalizations”, J. Math. Phys., 12 (1971), 498–501 | DOI | MR | Zbl

[25] Madore J., “Cosmological applications of the Lanczos Lagrangian”, Classical Quantum Gravity, 3 (1986), 361–371 | DOI | MR | Zbl

[26] Muto Y., “Critical Riemannian metrics”, Tensor, 29 (1975), 125–133 | MR | Zbl

[27] Nasu T., “On conformal invariants of higher order”, Hiroshima Math. J., 5 (1975), 43–60 | MR | Zbl

[28] Nomizu K., “On the decomposition of generalized curvature tensor fields. Codazzi, Ricci, Bianchi and Weyl revisited”, Differential Geometry, in Honor of Kentaro Yano, Kinokuniya, Tokyo, 1972, 335–345 | MR

[29] Patterson E. M., “A class of critical Riemannian metrics”, J. London Math. Soc., 2 (1981), 349–358 | DOI | MR | Zbl

[30] Reilly R. C., “Variational properties of functions of the mean curvatures for hypersurfaces in space forms”, J. Differential Geom., 8 (1973), 465–477 | MR | Zbl

[31] Reilly R. C., “Variational properties of mean curvatures”, Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical Congress, Vol. 2 (Dalhousie Univ., Halifax, N.S., 1971), 1971, 102–114 | MR

[32] Senovilla J. M. M., “Super-energy tensors”, Classical Quantum Gravity, 17 (2000), 2799–2842 ; gr-qc/9906087 | DOI | MR

[33] Sheng W., Trudinger N. S., Wang X.-J., “The Yamabe problem for higher order curvatures”, J. Differential Geom., 77:3 (2007), 515–553 | MR | Zbl

[34] Stehney A., “Courbure d'ordre $p$ et les classes de Pontrjagin”, J. Differential Geom., 8 (1973), 125–134 | MR | Zbl

[35] Thorpe J. A., “Some remarks on the Gauss–Bonnet integral”, J. Math. Mech., 18 (1969), 779–786 | MR | Zbl

[36] Viaclovsky J., “Conformal geometry and fully nonlinear equations”, Inspired by S. S. Chern, Nankai Tracts Math., 11, World Sci. Publ., Hackensack, NJ, 2006, 435–460 ; math.DG/0609158 | MR | Zbl

[37] Weyl H., “On the volume of tubes”, Amer. J. Math., 61 (1939), 461–472 | DOI | MR