Note on the Embedding of Manifolds in Euclidean Space

Low dimensional flat manifolds with some classes of Finsler metric

Flat Riemannian manifolds are (up to isometry) quotient spaces of the Euclidean space R^n over a Bieberbach group and there are an exact classification of of them in 2 and 3 dimensions. In this paper, two classes of flat Finslerian manifolds are stuided and classified in dimensions 2 and 3.

On the Quaternionic Curves in the Semi-Euclidean Space E_4_2

In this study, we investigate the semi-real quaternionic curves in the semi-Euclidean space E_4_2. Firstly, we introduce algebraic properties of semi-real quaternions. Then, we give some characterizations of semi-real quaternionic involute-evolute curves in the semi-Euclidean space E42 . Finally, we give an example illustrated with Mathematica Programme.

Learning Curved Manifolds The World is not always Flat or Learning Curved Manifolds

Manifold learning and finding low-dimensional structure in data is an important task. Many algorithms for this purpose embed data in Euclidean space, an approach which is destined to fail on non-flat data. This paper presents a non-iterative algebraic method for embedding the data into hyperbolic and spherical spaces. We argue that these spaces are often better than Euclidean space in capturing...

On the lower bounds for the number of periodic billiard trajectories in manifolds embedded in Euclidean space

KERNEL DENSITY ESTIMATION ON EMBEDDED MANIFOLDS WITH BOUNDARY By

We consider practical density estimation from large data sets sampled on manifolds embedded in Euclidean space. Existing density estimators on manifolds typically require prior knowledge of the geometry of the manifold, and all density estimation on embedded manifolds is restricted to compact manifolds without boundary. First, motivated by recent developments in kernel-based manifold learning, ...