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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)
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