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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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References:
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