Finsler型空间中的内蕴曲率和外内蕴曲率
Intrinsic and Extrinsic curvatures in Finsler-esque spaces
摘要：
我们考虑度量通过标量场$φ(x)$及其梯度$\nabla φ(x)$的泛函相互关联，并给出了与度量相关的一些关键几何量的变换.我们的分析提供了有用的和优雅的几何见解的作用，{ \it共形}和{ \it非共形}度量变形的内在和外在的几何$φ$ -叶理。作为一种特殊情况，我们比较了{ \it conformal}和{ \it disformal}变换，以突出一些非平凡的缩放差异。我们还研究了{ \it equi-geodesic}曲面的几何形成的点$p$在恒定的测地线距离$σ(p,P)$从一个固定的点$P$，并将我们的结果应用到一个特定的非正式几何的基础上$σ(p,P)$这是最近出现在一个最小长度的时空的背景下。
英文摘要：
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.
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