However, this question is about a straightforward embedding with no mapping (apologies if my terminology is off). I know there are different definitions of curvature and that a donut may be mapped to a 2-torus by, I assume, defining the angular distance as independent of the radius. Stretching would represent a space curvature, but the 2-torus is flat. To close it along the other dimension, I must connect the ends of the tube, but without stretching the paper. Class 6: Curved Space and Metrics In this class we will discuss the meaning of spatial curvature, how distances in a curved space can be measured using a metric, and how. This closes my 2-torus along one dimension. Now, if I try to construct a 2-torus, I first think of a flat sheet of paper rolled into a tube. If my thinking is correct, a 1-torus embedded in a 2D Euclidean space is just a circle (or any closed loop perhaps, but let's use a circle for simplicity).

jump to content jump to footer Institute for Parallel and Distributed Systems. Figure 2.3: The conformal embedding of global AdSd+1 into half of a (d+ 1)-. This thesis examines the impact of model hyperparameters, such as dimensionality and curvatures of the embedding space, on the embedding quality and examines the applicability of different hyperparameter optimization methods.

This actually makes a lot of sense, but also indicates that a donut may not be the best representation of a 2-torus. 3 Holographic Description of Curved-Space QFT and Multi-Layered AdS/CFT. However, in the linked question, a 3-torus is described more like 'an infinite mirrors cube' than anything like a curved donut. At a point with positive curvature the ball on the manifold will be smaller than the equivalent ball of radius in Euclidean space: a circle cut from a flat piece of paper will have to be folded or crumpled to cover the equivalent circle on the surface of a sphere. The model is an embedding diagram of the geometry of space near our sun. When we think of a torus shape in real life, we probably first think of a donut. Curvature in the space-space sheets produces no easily observed effect in the. This question is motivated by the previous one ( Can a flat space be closed?), although not directly related.

Curved space embedding