  We consider metrics related to each other by functionals of a scalar field
$\varphi(x)$ and it's gradient $\nabla \varphi(x)$, and give transformations of
some key geometric quantities associated with such metrics. Our analysis
provides useful and elegant geometric insights into the roles of {\it
conformal} and {\it non-conformal} metric deformations in terms of intrinsic
and extrinsic geometry of $\varphi$-foliations. As a special case, we compare
{\it conformal} and {\it disformal} transforms to highlight some non-trivial
scaling differences. We also study the geometry of {\it equi-geodesic} surfaces
formed by points $p$ at constant geodesic distance $\sigma(p,P)$ from a fixed
point $P$, and apply our results to a specific disformal geometry based on
$\sigma(p,P)$ which was recently shown to arise in the context of spacetime
with a minimal length.
