  We investigate the minimal and isoperimetric surface problems in a large
class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We
construct an adapted connection for such spaces and, using the variational
tools of Bryant, Griffiths and Grossman, derive succinct forms of the
Euler-Lagrange equations for critical points for the associated variational
problems. Using the Euler-Lagrange equations, we show that minimal and
isoperimetric surfaces satisfy a constant horizontal mean curvature conditions
away from characteristic points. Moreover, we use the formalism to construct a
horizontal second fundamental form, $II_0$, for vertically rigid spaces and, as
a first application, use $II_0$ to show that minimal surfaces cannot have
points of horizontal positive curvature and, that minimal surfaces in Carnot
groups cannot be locally horizontally geometrically convex. We note that the
convexity condition is distinct from others currently in the literature.
