Euclidean Domains A Euclidean domain is a nontrivial domain that is isomorphic to a Euclidean space. A Euclidean space is a vector space equipped with the s...
Euclidean Domains
A Euclidean domain is a nontrivial domain that is isomorphic to a Euclidean space. A Euclidean space is a vector space equipped with the structure of a metric space, which means it has a metric that induces the usual Euclidean distance on the vector space.
Euclidean domains are characterized by a few key properties, including:
They are non-singular, meaning that they have a finite basis of linearly independent vectors.
They are regular, meaning that they are the only domains that are isomorphic to Euclidean spaces.
They are geometrically simple, meaning that they can be drawn as the convex hull of a finite set of points.
Some important examples of Euclidean domains include:
Euclidean space itself.
The complex plane C^2.
The Grassmann manifold G(2,3).
Additional Notes
A Euclidean domain is a non-singular domain that is not isomorphic to any other Euclidean domain.
A Euclidean domain is a regular domain that is not isomorphic to any other domain that is not a Euclidean space.
A Euclidean domain is a geometrically simple domain that is not isomorphic to any other domain that is not a Euclidean space