I'm reading the classic paper "[Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1992__76_/PMIHES_1992__76__165_0/PMIHES_1992__76__165_0.pdf)" by Gromov-Schoen. In Section 6, they define the notion of F-connectedness as follows:



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> We say that a nonpositively curved complex $\mathrm{X}^k$ is F-*connected* if any two adjacent simplices are contained in a totally geodesic subcomplex $\mathrm{X}\_0$ which is isometric to a subset of the Euclidean space $\mathbb{R}^k$.
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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.



Understanding the definition of an F-connected simplicial complex
Here's a write-up of the comments, giving an example as asked for.


We assume that for two simplices in $X$ to be "adjacent" means that their intersection is nonempty. 


Let $X$ be the quotient of a pair of 2-simplices by identifying a vertex on one with a vertex on the other to a single vertex $V$. Each 2-simplex is given the metric of the standard 2-simplex (an equilateral triangle of side length 1). This is extended to a geodesic metric on $X$: the distance between a pair of points $x$ in one 2-simplex and $y$ in the other is the sum of the distances from $x$ and $y$ to $V$. 


It follows that for sufficiently small $r$, the sphere around $V$ of radius $r$ contains two components $I$ and $J$, each an interval, with the property that for each $x \in I$ and $y \in J$ we have $d(x,y)=r$. This does not happen for any subsets of Euclidean space with the Euclidean metric.


