The notion of generalized Berwald manifolds goes back to V. Wagner [60]. They are Finsler manifolds admitting linear connections on the base manifold such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). https://www.roneverhart.com/Craggy-Range-Sauvignon-Blanc-Te-Muna-Road-Vineyard-Martinborough-2022/