Abstract
The purpose of this paper is to present a geometrization of the Berger-Ebin theorem. We use this theorem for the study of harmonic maps and flows, in particular, Ricci solitons. We also clarify the role of a vector field in the corresponding decompositions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Berger and D. Ebin, “Some Decompositions of Space of Symmetric Tensors on a Riemannian Manifold,” J. Differential Geometry 3, 379–392 (1969).
A. Besse, Einstein Manifolds (Springer, Berlin, 1987; Mir, Moscow, 1990).
K. Yano, Differential Geometry on Complex and Almost Complex Spaces (Pergamon Press, Oxford, 1965).
R. S. Palais, Seminar on the Atiyah-Singer Index Theorem (Princeton, New Jersey, 1965; Mir, Moscow, 1970).
M. V. Smol’nikova, “On the Global Geometry of Harmonic Symmetric Bilinear Differential Forms,” Trudy MIAN 236, 328–331 (2002).
M. V. Smol’nikova, S. E. Stepanov, and I. G. Shandra, “Infinitesimal Harmonic Maps,” Izv. Vyssh. Uchebn. Zaved. Mat, No. 5, 69–75 (2004) [RussianMathematics (Iz. VUZ) 48 (5), 65–70 (2004)].
S. E. Stepanov and I. G. Shandra, “Geometry of Infinitesimal Harmonic Transformations,” Annals of Global Analysis and Geometry 24, 291–299 (2003).
N. K. Smolentsev “Spaces of Riemannian Metrics,” in Modern Mathematics and Its Applications: Geometry (Institute of Cybernetics, Georgia Academy of Sciences, 2005), Vol. 31, pp. 69–147.
T. Nore, “Second Fundamental Form of a Map,” Ann. Mat. Pura ed appl. 146, 281–310 (1987).
S. E. Stepanov, “On the Global Theory of Some Classes of Mappings,” Annals of Global Analysis and Geometry, 13, 239–249 (1995).
S. E. Stepanov, “On the Theory of Mappings of Riemannian Manifolds in the Large,” Izv. Vyssh. Uchebn. Zaved.Mat., No. 10, 81–88 (1994) [RussianMathematics (Iz. VUZ) 38 (10), 77–84 (1994)].
J. Davidov and A. G. Sergeev, “Twistor Spaces and Harmonic Maps,” Usp. Mat. Nauk 48(3), 3–96 (1993).
O. Nuhaud, “Transformations Infinitesimals Harmoniques,” C. R. Acad., Paris, Ser. A 274, 573–576 (1972).
R. S. Hamilton, “The Ricci Flow on Surfaces,” Contemp. Math. 71, 237–261 (1988).
S. E. Stepanov and V. N. Shelepova, “Note on Ricci Solitons,” Matem. Zametki 86(3), 474–477 (2009).
M. Eminenti, G. La Nave, and C. Mantegazza, “Ricci Solitons-the Equation Point of View,” Manuscripta Math. 127, 345–367 (2008).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S.E. Stepanov, 2012, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2012, No. 4, pp. 84–89.
About this article
Cite this article
Stepanov, S.E. The Berger-Ebin theorem and harmonic maps and flows. Russ Math. 56, 70–74 (2012). https://doi.org/10.3103/S1066369X12040093
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X12040093