This is a part extracted from a textbook (book "Riemann-Finsler geometry" by Chern & Shen):
(source: picofile.com)
.
My question: Why do we say that the tensor $\mathcal{J}$ (mean Landsberg tensor) is a mean of the Landsberg tensor $\mathcal{L}$? What is meaning of word "mean" here? Could somebody please help me to understand this?
The word "mean" here means the same as in "geometric mean" or "arithmetic mean". The (usual) Landsberg curvature measures the rate of changes of the Cartan torsion along geodesics. The "mean" Landsberg curvature is defined for the mean of $L$, which is given by $$ J_y(u):=\sum_{i,j=1}^n g^{ij}(y)L_y(b_i,b_j), $$ with the notation used in this reference. The family $J=(J_y)$ is then the "mean Landsberg curvature".