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Uniqueness theorem for Ricci tensor; compact and complete Riemannian manifolds; vanishing theorem
In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the following celebrated result: the Ricci curvature uniquely determines the Levi-Civita connection on any compact Einstein manifold with non-negative section curvature. In the present paper we generalize the result of DeTurck and Koiso for a Riemannian manifold with non-negative section curvature. In addition, we extended our result to complete non-compact Riemannian manifolds with nonnegative sectional curvature and with finite total scalar curvature.
[1] DeTurck, D., Koiso, N.: Uniqueness and non-existence of metrics with prescribed Ricci curvature. Annales de l’Institut Henri Poincare (C) Analyse non lineaire 1, 5 (1984), 351–359. MR 0779873 | Zbl 0556.53026
[2] Hamilton, R. S.: The Ricci curvature equation. Lecture notes: Seminar on nonlinear partial differential equations, Mathematical Sciences Research Institute Publications, Berkeley, 1983, 47–72. MR 0765228
[3] Becce, A. L.: Einstein manifolds. Springer-Verlag, Berlin–Heidelberg, 1987. MR 0867684
[4] Eells, J., Sampson, J. H.: Harmonic mappings of Riemannian manifolds. American Journal of Mathematics 86, 1 (1964), 109–160. DOI 10.2307/2373037 | MR 0164306 | Zbl 0122.40102
[5] Stepanov, S., Tsyganok, I.: Vanishing theorems for projective and harmonic mappings. Journal of Geometry 106, 3 (2015), 640–641. MR 1878047
[6] Yano, K., Bochner, S.: Curvature and Betti numbers. Princeton Univ. Press, Princeton, 1953. MR 0062505 | Zbl 0051.39402
[7] Vilms, J.: Totally geodesic maps. Journal of Differential Geometry 4, 1 (1970), 73–79. DOI 10.4310/jdg/1214429276 | MR 0262984 | Zbl 0194.52901
[8] Schoen, R., Yau, S. T.: Harmonic maps and topology of stable hypersurfaces and manifolds with non-negative Ricci curvature. Commenttarii Mathematici Helvetici 51, 1 (1976), 333–341. DOI 10.1007/BF02568161 | MR 0438388
[9] Yau, S. T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana University Mathematics Journal 25, 7 (1976), 659–679. DOI 10.1512/iumj.1976.25.25051 | MR 0417452 | Zbl 0335.53041
[10] Yau, S. T.: Seminar on Differential Geometry. Annals of Mathematics Studies, 102, Princeton Univ. Press, Princeton, NJ, 1982. MR 0645728 | Zbl 0471.00020
[11] Berger, M., Ebin, D.: Some decompositions of the space of symmetric tensors on a Riemannian manifold. Journal of Differential Geometry 3, 3-4 (1969), 379–392. DOI 10.4310/jdg/1214429060 | MR 0266084
[12] Pigola, S., Rigoli, M., Setti, A. G.: Vanishing and finiteness results in geometric analysis. A generalization of the Bochner technique. Birkhäuser, Basel, 2008. MR 2401291 | Zbl 1150.53001
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