Is our universe a 3 manifold?
Presumably our physical universe is globally a geometric 3-manifold. [A geometric 3-manifold is a space in which each point in the universe has a neighborhood which is isometric with a neighborhood of either Euclidean 3-space, a 3-sphere, or a hyperbolic 3-space.]
How many 3-manifolds are there?
Amazingly, every compact 2-manifold is homeomorphic to either a sphere (orientable), a connected sum of tori (orientable), or a connected sum of projective planes (nonorientable). There are infinitely many 3-manifolds.
What is toroidal geometry?
A toroid is a geometric shape that resembles a torus. A toroid is constructed by rotating a geometrical shape around an axis which is outside the shape. If this is done to a circle, a torus results.
What is a manifold in space?
A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in. ). To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round.
Is a torus manifold?
In topology, a ring torus is homeomorphic to the Cartesian product of two circles: S1 × S1, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. In the field of topology, a torus is any topological space that is homeomorphic to a torus.
What manifold is the universe?
geodesic manifold
The universe is often taken to be a geodesic manifold, free of topological defects; relaxing either of these complicates the analysis considerably. A global geometry is a local geometry plus a topology.
What is the use of toroid?
Toroidal inductors and transformers are used in a wide range of electronic circuits: power supplies, inverters, and amplifiers, which in turn are used in the vast majority of electrical equipment: TVs, radios, computers, and audio systems.
What are toroidal transformers used for?
Toroidal transformers are used in electronic applications that step up or down a voltage or for the isolation of electronic equipment from a source of voltage. Different transformers are used for different applications.
What exactly is a manifold?
The more typical definition is that a manifold is a space that is locally like Rn. To formalize it, you can say that a manifold is a space in which any sufficiently small neighborhood is homeomorphic to an open set in Rn. Homeomorphic means that you can find a homeomorphism between them.