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arXiv · 2603.18416

On the Finsler variational nature of autoparallels in metric-affine geometry

Lehel Csillag·Nicoleta Voicu·Salah Elgendi·Christian Pfeifer
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Abstract

In metric-affine geometry, autoparallels are generically non-variational, i.e., they are not the extremals of any action integral. The existence of a parametrization-invariant action principle for autoparallels is a long-standing open problem, which is equivalent to the so-called Finsler metrizability of the connection -- that is, to the fact that these autoparallels can be interpreted as Finsler geodesics. In this article, we address this problem for the class of torsion-free affine connections with vectorial nonmetricity, which includes, as notable subcases, Weyl and Schr\"odinger connections. For this class, we determine the necessary and sufficient conditions for the existence of a Finsler Lagrangian that metrizes the connection (and depends only algebraically on the metric and on the nonmetricity defining vector field). In the cases where such a Finsler Lagrangian exists, we construct it explicitly. In particular, we show that a broad class of such connections is in fact Finsler metrizable, i.e., the autoparallels of these connections are Finsler geodesics.

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Lehel Csillag, Nicoleta Voicu, Salah Elgendi, Christian Pfeifer. 2026-03-19. On the Finsler variational nature of autoparallels in metric-affine geometry. https://arxiv.org/abs/2603.18416

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