I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics and so on.

The matrix group I am studying is $UT(n)$, the $n\times n$ upper triangular matrices $\Pi=(\Pi_{i j})$ with positive diagonal elements and $\Pi_{11}=1$; but specially $UT(3)$.

The metric is defined in the book of C.G. Small *The Statistical Theory of Shape* ( p. 101-106):
Let $\Pi_x$ and $\Pi_{x+dx}$ in $UT(n)$
and let $\Lambda = \Pi_{x+dx}\cdot \Pi_{x}^{-1}$. The infinitesimal
distance ds from $\Pi_x$ to $\Pi_{x+dx}$ is given by the formula
$$ds^2= \frac{\sum (\lambda_j-\bar\lambda)^2}{n}$$ where $\lambda_j$ are the eigenvalues of $\Lambda^T\Lambda$ and $\bar\lambda=\frac {\sum \lambda_j}{n}$.

Looking for Lie groups I found references for metric questions, but in a very general setting of differential geometry.

In the other hand, I found introduction references for matrix groups, but they avoid metric considerations.

And thus my question: Are there references in matrix groups that avoid generalities but have a lot of results in the metric structure?