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(topological space) Manifold vs Euclidean space

highmath

Member
Sep 24, 2018
36
What is the difference between Manifold space to Euclidean space?

What properties does the Manifold space have that Euclidean space doesn't have?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,643
Leiden
What is the difference between Manifold space to Euclidean space?

What properties does the Manifold space have that Euclidean space doesn't have?
A manifold is an abstraction of Euclidean space, so it has less properties, and allows for more.
That is, until we add extra properties to a manifold that are not possible in Euclidean space.

A manifold is only locally homeomorphic (has a bi continuous function) to Euclidean space $\mathbb R^n$.
Homeomorphic means that there is a continuous 1-1 mapping that has an inverse that is also continuous.
It means that a manifold does have angles and distances, but these can change from point to point in a continuous fashion.
As such it allows:
  • Curved spaces since the global parallel postulate of Euclidean geometry is not (necessarily) true.
  • Non-orientable shapes like the Klein bottle.
  • Disconnected components that may even have different dimensions.