This dissertation addresses some investigations of Finsler geometry and homogeneous Riemannian nilmanifolds. Firstly, we study a class of Finsler metric that contains the class of Berwald metric. Then, we investigate the relationships among the classes obtained. This will enhance the understanding of the role of the relevant tensors in characterizing the new classes of Finsler metrics. Finally, we describe the sets of the geodesic vectors and the totally geodesic subalgebras of the six-dimensional filiform nilmanifolds. In this class, with the exception of the metric Lie algebras corresponding to the standard filiform Lie algebra, the flat totally geodesic subalgebras of every metric Lie algebra have a dimension of at most two.