Intrinsic and extrinsic curvatures in Finsler esque spaces

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