Definition of a great circle in $S^n$ I can't seem to find a nice definition, the only things I found are "intersection of $S^n$ with a two-dimensional sub-space" and "separates $S^n$ in two parts of equal volume".
Isn't there something with more symbols? Or is that it?
 A: You can also define the great circle as geodetic line of $S^n$. I'm not sure if the geodetic equation satisfies your quest of more symbols, though.
Another way to define the great circle is as the circle on $S^n$ with largest perimeter.
Yet another way to define it is to construct a descending chain of spheres:
Start with a point and its antipodal point (the point of largest distance to the initial point). Then the set of all points that have equal distance to both the original point and its antipodal forms a maximal $S^{n-1}$. On that $S^{n-1}$ you can do the same construction to arrive at a maximal $S^{n-2}$, and then continue recursively until you reach an $S^1$. That $S^1$ will then be a great circle of the $S^n$ you started with. Formalizing this process should produce quite a few symbols.
Note that you can also define the end of that process by noticing that if you continue doing the same with the $S^1$ you get a disconnected set (namely two antipodal points), and therefore the great circle is the last connected set you get in that sequence.
