Abstract
The Ricci flow is an evolution equation in the space of Riemannian metrics.A solution for this equation is a curve on the manifold of Riemannian metrics. In this paper we introduce a metric on the manifold of Riemannian metrics such that the Ricci flow becomes a geodesic.We show that the Ricci solitons introduce a special slice on the manifold of Riemannian metrics.
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References
B. Andrews, A. Hopper, “The Ricci Flow in Riemannian Geometry: A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem”, Lecture Notes in Mathematics, 2011, 2011.
M. Bauer, P. Harms, P. W. Michor, “Sobolev metrics on the manifold of all riemannian metrics”, J. Differ. Geom., 94(2), 187–365, 2013.
M. Bauer, P. Harms, P. W. Michor, “Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation”, Ann. Glob. Anal. Geom., 41(4), 461–472, 2012.
B. Clarke, “The MetricGeometry of the Manifold of Riemannian Metrics over a Closed Manifold”, Calc.Var., 39, 533–545, 2010.
B. Clarke, “The Completion of the Manifold of Riemannian Metrics” J. Differ. Geom., 93(2), 203–268, 2013.
A. S. Dancer, M. Y. Wang, “On Ricci solitons of cohomogeneity one”, Ann. Glob. Anal. Geom, 39(3), 259–292, 2011.
D. M. DeTurck, “Deforming metrics in the direction of their Ricci tensors”, J. Differ. Geom., 18(1), 157–162, 1983.
D. Ebin, “The manifold of Riemannian metrics”, Proc.Symp. PureMath., 15, 11–40, 1970.
J. Eells, J. H. Sampson, “Harmonic map**s of Riemannian manifolds”, AM. J. MATH., 86(1), 109–160, 1964.
D. S. Freed, D. Groisser, “The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group”, Michigan Math. J., 36, 323–344, 1989.
H. Ghahremani-Gol, A. Razavi, “Ricci flow and the manifold of Riemannian metrics”, Balkan J.Geom. Appl., 18(2), 20–30, 2013.
O. Gil-Medrano, P. W. Michor, “The Riemannian manifold of all Riemannian metrics”, Q. J. Math.Oxf. Ser. (2) 42(166), 183–202, 1991.
R. S. Hamilton, “The inverse function theorem of Nash and Moser”, Bull. Am. Math. Soc., 7(1), 65–222, 1982.
R. S. Hamilton, “Three-manifolds with positive Ricci curvature”, J. Differ. Geom., 17(2), 255–306, 1982.
A. Kriegl, P. W. Michor, The convenient setting of global analysis (volume 53 ofMathematical Surveys and Monographs, AmericanMathematical Society, Providence, 1997).
P. W. Michor, D. Mumford, “Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms”, Doc. Math., 10, 217–245, 2005.
P. W. Michor, D. Mumford, “Riemannian geometries on spaces of plane curves”, J. Eur. Math. Soc. (JEMS), 8, 1–48, 2006.
M. Nitta, “Conformal sigma models with anomalous dimensions and Ricci solitons”, Mod. Phys. Lett. A, 20, 577–584, 2005.
D. Perrone, “Geodesic Ricci solitons on unit tangent sphere bundles”, Ann. Glob. Anal. Geom, 44(2), 91–103, 2013.
N. K. Smolentsev, “Natural weak Riemannian structures on the space of Riemannian metrics”, Siberian Math J., 35(2), 396–402, 1994.
E. Tsatis, “Mean curvature flow on Ricci solitons”, J. Phys. A: Math. Theor., 43, 045202–045215, 2010.
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Original Russian Text © H. Ghahremani-Gol, A. Razavi, 2016, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2016, No. 5, pp. 38-48.
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Ghahremani-Gol, H., Razavi, A. The Ricci flow as a geodesic on the manifold of Riemannian metrics. J. Contemp. Mathemat. Anal. 51, 215–221 (2016). https://doi.org/10.3103/S1068362316050010
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DOI: https://doi.org/10.3103/S1068362316050010