Intrinsic and Extrinsic curvatures in Finsler-esque spaces

Abstract

We consider metrics related to each other by functionals of a scalar field $\phi (x)$ and it's gradient $\nabla \phi (x)$, 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 $\phi$-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.

8 pages, 1 figure; Sec IV now includes a more refined expression for Ricci scalar completely in terms of conformal (3-)geometry, and discusses a couple of applications; to appear in Gen. Rel. Grav._

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