Abstract
We illustrate some well-known facts about the evolution of the 3-sphere (S 3, g) generated by the Ricci flow. We define the Dirac flow and study the properties of the metric \(\bar g = dt^2 + g(t)\), where g(t) is a solution of the Dirac flow. In the case of a metric g conformally equivalent to the round metric on S 3 the metric \(\bar g\) is of constant curvature. We study the properties of solutions in the case when g depends on two functional parameters. The flow on differential 1-forms whose solution generates the Eguchi–Hanson metric was written down. In particular cases we study the singularities developed by these flows.
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References
Onda K., “The Ricci flow on 3-dimensional Lie groups and 4-dimensional Ricci-flat manifolds,” Adv. Appl. Math. Sci., 11, No. 3, 133–159 (2012).
Kobayashi Sh. and Nomizu K., Foundations of Differential Geometry. 2 vols, Interscience Publishers, New York; London (1963).
Besse A. L., Einstein Manifolds, Springer-Verlag, Berlin (2008).
Gibbons G. W., Hawking S. W., “Classification of gravitational instanton symmetries,” Comm. Math. Phys., 66, No. 3, 291–310 (1979).
Eguchi T. and Hanson A. J., “Self-dual solutions to Euclidean gravity,” Ann. Phys., 120, No. 1, 82–106 (1979).
Hitchin N., “Special holonomy and beyond,” in: Strings and Geometry, Amer. Math. Soc., Providence, 2004, pp. 159–175 (Clay Math. Proc.; V. 3).
Bazaĭkin Ya. V. and Malkovich E. G., “Spin(7)-structures on complex linear bundles and explicit Riemannian metrics with holonomy group SU(4),” Sb. Math., 202, No. 4, 467–493 (2011).
Bazaĭkin Ya. V., “On the new examples of complete noncompact Spin(7)-holonomy metrics,” Sib. Math. J., 48, No. 1, 8–25 (2007).
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The author was supported by the Government of the Russian Federation (Grant 14.B25.31.0029).
Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 2, pp. 432–446, March–April, 2016; DOI: 10.17377/smzh.2016.57.216.
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Malkovich, E.G. Dirac flow on the 3-sphere. Sib Math J 57, 340–351 (2016). https://doi.org/10.1134/S0037446616020166
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DOI: https://doi.org/10.1134/S0037446616020166