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kent davidge
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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.kent davidge said: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##?
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.
Bijectivity of manifolds refers to the concept that there is a one-to-one correspondence between points on two different manifolds. This is important because it allows us to understand the topological structure of the manifolds and how they relate to each other.
Yes, it is possible for two manifolds to have different dimensions and still be bijective. This is because bijectivity only requires a one-to-one correspondence between points, not necessarily the same number of points.
Bijectivity has a significant impact on the geometry of manifolds. It allows us to understand the relationship between different manifolds in terms of their shape and structure. It also helps us to identify symmetries and transformations between manifolds.
There are some restrictions on the types of manifolds that can be bijective. For example, they must be smooth and continuous, and they cannot have any holes or self-intersections. Additionally, they must have the same dimension.
Bijectivity of manifolds is closely related to other mathematical concepts, such as homeomorphism, diffeomorphism, and isomorphism. These all involve a one-to-one correspondence between points, but differ in terms of the specific properties being preserved.