I Bijectivity of Manifolds: Can m≠n?

[kent davidge]

Is it possible for a manifold to be homeomorphic to $R^m$ in some regions and homeomorphic to $R^n$ in other regions, with $m \neq n$?

 

[fresh_42]

Is it possible for a manifold to be homeomorphic to $R^m$ in some regions and homeomorphic to $R^n$ in other regions, with $m \neq n$?

This depends on how you define a manifold. Let's say that we only assume it to be locally Euclidean. Then the two parts cannot be connected, as this would lead to a point, where it isn't Euclidean anymore. But if we allow more than one component, then it is possible, although we would probably investigate each of the components on their own.

 

[Matterwave]

To just expand on fresh_42 a little, I quote from wikipedia:

Generally manifolds are taken to have a fixed dimension (the space must be locally homeomorphic to a fixed n-ball), and such a space is called an n-manifold; however, some authors admit manifolds where different points can have different dimensions.[1] If a manifold has a fixed dimension, it is called a pure manifold. For example, the sphere has a constant dimension of 2 and is therefore a pure manifold whereas the disjoint union of a sphere and a line in three-dimensional space is not a pure manifold. Since dimension is a local invariant (i.e. the map sending each point to the dimension of its neighbourhood over which a chart is defined, is locally constant), each connected component has a fixed dimension.

Since this question was posed in the physics (SR/GR) sub-forum, I will specify that generally we consider only a "pure manifold" when doing physics. This is because if you have a disjoint union of two sets as your "manifold", then those disjoint unions can't actually interact with each other. There's no path that takes you from one to the other. You can't send signals between the two, and so, for observers in one manifold, the other manifold might as well not exist.