A metric basis characterization of Euclidean space

A Metric Characterization of Snowflakes of Euclidean Spaces

We give a metric characterization of snowflakes of Euclidean spaces. Namely, a metric space is isometric to Rn equipped with a distance (dE) , for some n ∈ N0 and ∈ (0, 1], where dE is the Euclidean distance, if and only if it is locally compact, 2-point isometrically homogeneous, and admits dilations of any factor.

A Menger Redux: Embedding Metric Spaces Isometrically in Euclidean Space

We present geometric proofs of Menger’s results on isometrically embedding metric spaces in Euclidean space. In 1928, Karl Menger [6] published the proof of a beautiful characterization of those metric spaces that are isometrically embeddable in the ndimensional Euclidean space E. While a visitor at Harvard University and the Rice Institute in Houston during the 1930-31 academic year, Menger ga...

Tangent Bundle of the Hypersurfaces in a Euclidean Space

Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper...

Convexity and the Euclidean Metric of Space-Time

We address the reasons why the “Wick-rotated”, positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point o...

A nonstandard analysis characterization of submanifolds in Euclidean space