  We show that for a surface S, the subgraph of the pants graph determined by
fixing a collection of curves that cut S into pairs of pants, once-punctured
tori, and four-times-punctured spheres is totally geodesic. The main theorem
resolves a special case of a conjecture made by Aramayona, Parlier, and
Shackleton and has the implication that an embedded product of Farey graphs in
any pants graph is totally geodesic. In addition, we show that a pants graph
contains a convex n-flat if and only if it contains an n-quasi-flat.
