Intrinsic and Extrinsic curvatures in Finsler-esque spaces

TL;DR

This paper explores the geometric effects of conformal and disformal metric transformations related to scalar fields, analyzing intrinsic and extrinsic curvature changes, and applies these insights to minimal length spacetime geometries.

Contribution

It provides a detailed analysis of how conformal and disformal transformations affect geometric quantities and introduces a study of equi-geodesic surfaces in disformal geometries.

Findings

Conformal and disformal transformations have distinct impacts on curvature.

Geometric properties of equi-geodesic surfaces are characterized.

Disformal geometry based on geodesic distance relates to minimal length spacetime models.

Abstract

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.