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Manifold; linear connection; metric connection; pseudo-Riemannian geometry
We contribute to the reverse of the Fundamental Theorem of Riemannian geometry: if a symmetric linear connection on a manifold is given, find non-degenerate metrics compatible with the connection (locally or globally) if there are any. The problem is not easy in general. For nowhere flat $2$-manifolds, we formulate necessary and sufficient metrizability conditions. In the favourable case, we describe all compatible metrics in terms of the Ricci tensor. We propose an application in the calculus of variations.
[1] Boothby, W. M.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, Amsterdam–London–New York–Oxford–Paris–Tokyo, 2003 (revised second editon). MR 0861409
[2] Cocos, M.: A note on symmetric connections. J. Geom. Phys. 56 (2006), 337–343. MR 2171888 | Zbl 1091.53008
[3] do Carmo, M. P.: Riemannian Geometry. Birkhäuser, Boston–Basel–Berlin, 1992. MR 1138207 | Zbl 0752.53001
[4] Cheng, K. S, Ni, W. T.: Necessary and sufficient conditions for the existence of metrics in two-dimensional affine manifolds. Chinese J. Phys. 16 (1978), 228–232.
[5] Douglas, J.: Solution of the inverse problem of the calculus of variations. Trans. AMS 50 (1941), 71–128. MR 0004740 | Zbl 0025.18102
[6] Dodson, C. T. J., Poston, T.: Tensor Geometry. The Geometric Viewpoint and its Uses. Spriger, New York–Berlin–Heidelberg, 1991 (second editon). MR 1223091 | Zbl 0732.53002
[7] Eisenhart, L. P., Veblen, O.: The Riemann geometry and its generalization. Proc. London Math. Soc. 8 (1922), 19–23.
[8] Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin–Heidelberg–New York, 2005. MR 2165400 | Zbl 1083.53001
[9] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry I, II. Wiley, New York–Chichester–Brisbane–Toronto–Singapore, 1991.
[10] Kolář, I., Slovák, J., Michor, P. W.: Natural Operations in Differential Geometry. Springer, Berlin–Heidelberg–New York, 1993. MR 1202431
[11] Kowalski, O.: On regular curvature structures. Math. Z. 125 (1972), 129–138. MR 0295250 | Zbl 0234.53024
[12] Lovelock, D., Rund, H.: Tensors, Differential Forms, and Variational Principle. Wiley, New York–London–Sydney, 1975. MR 0474046
[13] Mikeš, J., Kiosak, V., Vanžurová, A.: Geodesic Mappings of Manifolds with Affine Connection. Palacký Univ. Publ., Olomouc, 2008. MR 2488821 | Zbl 1176.53004
[14] Nomizu, K., Sasaki, T.: Affine Differential Geometry. Geometry of Affine Immersions. Cambridge Univ. Press, Cambridge, 1994. MR 1311248
[15] Petrov, A. Z.: Einstein Spaces. Moscow, 1961 (in Russian). MR 0141492
[16] Schmidt, B. G.: Conditions on a connection to be a metric connection. Commun. Math. Phys. 29 (1973), 55–59. MR 0322726
[17] Sinyukov, N. S.: Geodesic Mappings of Riemannian Spaces. Moscow, 1979 (in Russian). MR 0552022 | Zbl 0637.53020
[18] Thompson, G.: Local and global existence of metrics in two-dimensional affine manifolds. Chinese J. Phys. 19, 6 (1991), 529–532.
[19] Vanžurová, A.: Linear connections on two-manifolds and SODEs. Proc. Conf. Aplimat 2007, Bratislava, Slov. Rep., Part II, 2007, 325–332.
[20] Vanžurová, A.: Metrization problem for linear connections and holonomy algebras. Archivum Mathematicum (Brno) 44 (2008), 339–348. MR 2501581
[21] Vanžurová, A.: Metrization of linear connections, holonomy groups and holonomy algebras. Acta Physica Debrecina 42 (2008), 39–48.
[22] Vanžurová, A., Žáčková, P.: Metrization of linear connections. Aplimat, J. of Applied Math. (Bratislava) 2, 1 (2009), 151–163.
[23] Wolf, J. A.: Spaces of Constant Curvature. Berkley, California, 1972. MR 0343213
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