For torsion-free connections with vectorial nonmetricity, necessary and sufficient conditions are derived for Finsler metrizability, with explicit Lagrangian construction when the conditions hold.
Berwald m-kropina spaces of arbitrary signature: Metrizability and ricci-flatness.Journal of Mathematical Physics, 65(12):122502, 12 2024
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On the Finsler variational nature of autoparallels in metric-affine geometry
For torsion-free connections with vectorial nonmetricity, necessary and sufficient conditions are derived for Finsler metrizability, with explicit Lagrangian construction when the conditions hold.