## Metrizability of connections on two-manifolds.(English)Zbl 1195.53023

A manifold $$M$$ equipped with a linear connection $$\nabla$$ is said to be metrizable if there exists a metric compatible with $$\nabla$$. In particular, if $$\nabla$$ is symmetric, a solution of the metrization problem is a metric $$g$$ such that $$\nabla$$ is just the Levi-Cività connection of $$(M, g)$$.
The authors discuss the metrization problem for 2-manifolds.
In particular, they prove that, given a nowhere flat symmetric connection $$\nabla$$ on a 2-dimensional manifold, then $$\nabla$$ is metrizable if and only if its Ricci tensor Ric is regular, symmetric and there exists a smooth function $$f$$ such that $$\nabla\text{\,Ric}= df\otimes\text{Ric}$$. Moreover, when $$\nabla$$ is metrizable, all the compatible metrics set up the 1-parameter family described by $$g_\lambda= \exp(- f+ t)\text{Ric}$$, $$t\in\mathbb{R}$$.
Furthermore, the authors point out the relationship of the metrization problem with the so-called Inverse Problem of the calculus of variations and provide several examples.

### MSC:

 53B05 Linear and affine connections 53B20 Local Riemannian geometry 53C05 Connections (general theory)
Full Text:

### References:

 [1] Boothby, W. M.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, Amsterdam-London-New York-Oxford-Paris-Tokyo, 2003 · Zbl 0333.53001 [2] Cocos, M.: A note on symmetric connections. J. Geom. Phys. 56 (2006), 337-343. · Zbl 1091.53008 [3] do Carmo, M. P.: Riemannian Geometry. Birkhäuser, Boston-Basel-Berlin, 1992. · 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. · Zbl 0025.18102 [6] Dodson, C. T. J., Poston, T.: Tensor Geometry. The Geometric Viewpoint and its Uses. Spriger, New York-Berlin-Heidelberg, 1991 · Zbl 0732.53002 [7] Eisenhart, L. P., Veblen, O.: The Riemann geometry and its generalization. Proc. London Math. Soc. 8 (1922), 19-23. · JFM 48.0842.02 [8] Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin-Heidelberg-New York, 2005. · Zbl 1083.53001 [9] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry I, II. Wiley, New York-Chichester-Brisbane-Toronto-Singapore, 1991. · Zbl 0119.37502 [10] Kolář, I., Slovák, J., Michor, P. W.: Natural Operations in Differential Geometry. Springer, Berlin-Heidelberg-New York, 1993. · Zbl 0782.53013 [11] Kowalski, O.: On regular curvature structures. Math. Z. 125 (1972), 129-138. · Zbl 0234.53024 [12] Lovelock, D., Rund, H.: Tensors, Differential Forms, and Variational Principle. Wiley, New York-London-Sydney, 1975. · Zbl 0308.53008 [13] Mikeš, J., Kiosak, V., Vanžurová, A.: Geodesic Mappings of Manifolds with Affine Connection. Palacký Univ. Publ., Olomouc, 2008. · Zbl 1176.53004 [14] Nomizu, K., Sasaki, T.: Affine Differential Geometry. Geometry of Affine Immersions. Cambridge Univ. Press, Cambridge, 1994. · Zbl 0834.53002 [15] Petrov, A. Z.: Einstein Spaces. Moscow, 1961 · Zbl 0098.18901 [16] Schmidt, B. G.: Conditions on a connection to be a metric connection. Commun. Math. Phys. 29 (1973), 55-59. · Zbl 0243.53019 [17] Sinyukov, N. S.: Geodesic Mappings of Riemannian Spaces. Moscow, 1979 · 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. · Zbl 1212.53021 [21] Vanžurová, A.: Metrization of linear connections, holonomy groups and holonomy algebras. Acta Physica Debrecina 42 (2008), 39-48. · Zbl 1212.53021 [22] Vanžurová, A., Žáčková, P.: Metrization of linear connections. Aplimat, J. of Applied Math. (Bratislava) 2, 1 (2009), 151-163. · Zbl 1212.53020 [23] Wolf, J. A.: Spaces of Constant Curvature. Berkley, California, 1972.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.