On some geometrical properties of six-dimensional nilpotent Lie group $\mathcal{N}_{6,11}$

In this paper, we study left invariant Randers metric $F$ on six-dimensional nilpotent Lie group $\mathcal{N}_{6,11}$ with Lie algebra $\mathfrak{n}_{6,11}.$ We compute Levi-civita connection, Riemann curvature tensor, sectional curvature, Ricci curvature and scalar curvature on $(\mathcal{N}_{6,11},F)$. Moreover, we classify simply connected six-dimensional nilpotent Lie groups equipped with $(\alpha,\beta)$-metrics of Douglas and Berwald type defined by left-invariant Riemannian metric and left-invariant vector field. We also compute S-curvature and flag curvature. At the end it is also proved that $(\mathcal{N}_{6,11},F)$ can neither be naturally reductive nor Ricci-quadratic.