Paper detail

Intrinsic and Extrinsic curvatures in Finsler-esque spaces

We consider metrics related to each other by functionals of a scalar field $φ(x)$ and it's gradient $\nabla φ(x)$, 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 $φ$-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 $σ(p,P)$ from a fixed point $P$, and apply our results to a specific disformal geometry based on $σ(p,P)$ which was recently shown to arise in the context of spacetime with a minimal length.