I'm reading the classic paper "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one" by Gromov-Schoen. In Section 6, they define the notion of F-connectedness as follows:

We say that a nonpositively curved complex $\mathrm{X}^k$ is F-

connectedif any two adjacent simplices are contained in a totally geodesic subcomplex $\mathrm{X}_0$ which is isometric to a subset of the Euclidean space $\mathbb{R}^k$.

Here, $k$ is the maximum dimension of the simplices in $\mathrm{X}^k$. I understand the notion of nonpositive curvature (in the Alexandrov sense), but I can't understand how F-connectedness places any further restriction on the geometry of the complex. For the entire paper, the standing assumption has been made that the complex is (locally) isometrically embedded in $\mathbb{R}^n$. It seems to me (and I'm sure this is wrong, but I'm not sure why) that an isometry can be cooked up between the union of any two adjacent simplices and some subset of $\mathbb{R}^k$--since the simplices are of dimension at most k--and that this union should itself be a totally geodesic subcomplex.

In particular, what would be an example of a non-positively curved complex that is not F-connected?

I feel like I'm missing something obvious but important.