Geometric Applications of the Berger–Ebin and York Orthogonal Decompositions of the Space of Symmetric Two-forms on a Compact Riemannian Manifold

Abstract

The Berger–Ebin and York 𝑳𝟐-orthogonal decompositions of symmetric bilinear differential two-forms on a compact Riemannian manifold are fundamental tools in global Riemannian geometry. In the present paper, we consider geometric interpretations and applications of these decompositions. Namely, we investigate the structure of Ricci tensors on compact Riemannian manifolds, with a particular focus on compact Ricci almost solitons, utilizing both the Berger–Ebin and York 𝑳𝟐-orthogonal decompositions. In addition, we explore applications of the York 𝑳𝟐-orthogonal decomposition to submanifold theory and use the Berger–Ebin 𝑳𝟐-orthogonal decomposition to study harmonic maps of compact Riemannian manifolds.

Keywords

• Compact Riemannian manifold
• vector space of differential two-forms
• Berger–Ebin and York $L^{2}$-orthogonal decompositions

References

1. Berger M., Ebin D.: Some decompositions of the space of symmetric tensors on a Riemannian manifold. J. Differential Geometry, 3, 379-392 (1969).
2. Besse A.L.: Einstein manifolds, Springer-Verlag, Berlin & Heidelberg (2008).
3. Bourguignon, J. P. , Ebin, D. G., Marsden J. E.: Sur le noyau des opérateurs pseudo-differentiels á symbole surjectif et non injectif. C R Acad Sci Paris Sér A-B, 282, A867–A870 (1976).
4. Bourguignon J.-P., Ezin J.-P.: Scalar Curvature Functions in a Conformal Class of Metrics and Conformal Transformations. Transactions of AMS, 301 (2), 623-736 (1987).
5. Branson T.: Stein-Weiss operators and ellipticity. Journal of Functional, 151, 334-383 (1997).
6. Carlotto A.: The general relativistic constraint equations. Living Reviews in Relativity, 24 (2), 1-170 (2021).
7. York J. W.: Covariant decompositions of symmetric tensors in the theory of gravitation. Ann. Inst. H. Poincar´e Sect. A (N.S.), 21 (4), 319–332 (1974).
8. Chow B., Lu P., Ni L.: Hamilton’s Ricci flow, in Grad. Stud. in Math., 77, AMS, Providence, RI (2006).

9. Chen B.-Y.: Riemannian Submanifolds, Handbook of Differential Geometry, North Holland, vol. I, 187–418 (2000).
10. Dahl M., Rkoncke K.: Local and global scalar curvature rigidity of Einstein manifolds. Mathematische Annalen, 388, 453–510 (2024).
11. Ebin D.: The manifold of Riemannian metrics. Proc. Symp. AMS, 15, 11–40 (1970).
12. Eells, J., Lemaire L.: A report on harmonic maps. Bull. London Math. Soc., 10, 1-68 (1978).
13. Fischer A.E., Marsden J.E.: The manifold of conformally equivalent metric. Can. J. Math., XXIX (1), 193-209 (1977).
14. Gicquaud R., Ngo Q.A.: A new point of view on the solutions to the Einstein constraint equations with arbitrary mean curvature and small TT-tensor. Class. Quant. Grav., 31 (19) (2014), 195014.
15. Kozlowski W., Pierzchalski A.: Natural boundary value problems for weighted form Laplacians. Ann. Scuola Norm. Sup. Pisa Cl. Sci., VII (5), 343-367 (2008).
16. Mazur P.O., Mottola E.: The path integral measure, conformal factor problem and stability of the group state of quantum gravity. Nuclear Physics, B341, 187-212 (1990).
17. Mikeš J., Stepanov S., Tsyganok I.: Application of the L2-orthogonal decomposition of the second fundamental form of the hypersurfaces and the Ahlfors Laplacian to the study of the general relativistic vacuum constraint equations. International Journal of Geometric Methods in Modern Physics 22(8), article number 2550044 (2025).
18. Morgan J., Tian G.: Ricci flow and Poincare conjecture, Clay Mathematics Monographs, 3. AMS, Providence, RI; Clay Mathematics Institute, Cambridge, MA (2007).
19. Nore T.: Second fundamental form of a map. Ann. Mat. Pura ed appl. 146, 281–310 (1987).
20. Pierzchalski A.: Gradients: the ellipticity and the elliptic boundary conditions – a jigsaw puzzle. Folia Mathematica, 19 (1), 65-83 (2017).
21. Pierzchalski A., Orsted B.: The Ahlfors Laplacian on a Riemannian manifold with boundary. Michigan Math. J., 43, 99-122 (1996).
22. Pigola S., Rigoli M., Rimoldi M., Setti A. G.: Ricci almost solitons. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (4), 757–799 (2011).
23. Rademacher H.-B.: Einstein spaces with a conformal group. Res. Math., 56 (1), 421–444 (2009).
24. Rovenski, V., Stepanov, S., Tsyganok, I.: The Bourguignon Laplacian and harmonic symmetric bilinear forms. Mathematics, 8 (1), article number 83 (2020).
25. Schoen R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom., 20 (2), 479-495 (1984).
26. Stepanov S.E., Shandra I.G.: Geometry of infinitesimal harmonic transformations. Ann. Glob. Anal. Geom., 24, 291-299 (2003).
27. Stepanov S.E.: On the global theory of some classes of mappings. Annals of Global Analysis and Geometry, 13, 239–249 (1995).
28. Stepanov S.E., Shandra I.G.: Harmonic diffeomorphisms of manifolds. St. Petersburg Math. J., 16, 401-412 (2005).
29. Yano K.: Integral formulas in Riemannian geometry, Marcel Dekker, New York, (1970).
30. Yano K., Ishihara S.: Harmonic and relatively affine mappings. Journal of Differential Geometry, 10, 501-509 (1975).