Complex sentence: Finsler geometry has the "Finsler manifold" as the main object of study--this is a differential manifold with a Finsler metric, i.e. a Banach norm defined on each tangent space.
Split into simpler sentences: Finsler geometry has the "Finsler manifold" as the main object of study. This is a differential manifold with a Finsler metric, i.e. a Banach norm defined on each tangent space.