We study $G$-almost geodesic mappings of the second type $\underset{\theta }{\to }{\pi }_2\left(e\right)$, $\theta =1,2$ between non-symmetric affine connection spaces. These mappings are a generalization of the second type almost geodesic mappings defined by N. S. Sinyukov (1979). We investigate a special type of these mappings in this paper. We also consider $e$-structures that generate mappings of type $\underset{\theta }{\to }{\pi }_2\left(e\right)$, $\theta =1,2$. For a mapping $\underset{\theta }{\to }{\pi }_2(e,F)$, $\theta =1,2$, we determine the basic equations which generate them.

Let ϕ :(M,F)→ (N,h) be a harmonic map from a Finsler manifold to any Riemannian manifold. We establish Bochner's formula for the energy density of ϕ and maximum principle on Finsler manifolds, from which we deduce some properties of harmonic maps ϕ.