Another manifold definition deficiency?

In summary, conventional manifold definition refers to the neighbor of every point having a Euclidean space description. However, most manifolds have an additional property of curvature, which means that the manifold definition is actually describing a tangent space, rather than a curved manifold. This requires the coordinate transformations on overlapping charts to be at least twice differentiable. The degree of differentiability of the coordinate transformations is an important factor in determining the compatibility with a C2 structure, which is necessary for defining curvature.
  • #1
zankaon
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Conventional manifold definition refers to the neighbor of every point having a Euclidean space description. http://en.wikipedia.org/wiki/Manifold" [Broken] But if most manifolds have additional property of some curvature, then won't such manifold definition actually be describing a tangent space i.e. not part of curved manifold?
 
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  • #2
zankaon said:
Conventional manifold definition refers to the neighbor of every point having a Euclidean space description.
Only the topology -- none of the other properties of Euclidean space are used. Note the fact that each chart is merely a homeomorphism -- an isomorphism of topologies -- as opposed to anything stronger like a diffeomorphism (isomorphism of differential calculus) or an isometry (isomorphism of geometry).

Manifolds are purely topological in nature -- they don't have any geometry or differential structure. Those are extra structure we might add in addition to being a manifold (e.g. differential manifolds and Riemannian manifolds)
 
  • #3
zankaon said:
Conventional manifold definition refers to the neighbor of every point having a Euclidean space description. http://en.wikipedia.org/wiki/Manifold" [Broken] But if most manifolds have additional property of some curvature, then won't such manifold definition actually be describing a tangent space i.e. not part of curved manifold?

Not withstanding homeomorphism definition of manifold, wherein one has 1:1 mapping and bicontinuity (attention to inbetweenness); might one also consider the conjecture (?) that perhaps most manifolds are non-smooth;that is, non-differential. Hence one could not consider the neighborhood of a 'point' nor a patch of manifold. So the concept of tangent space would not even be realizable. Also the word Euclidean, to most, implies the additional property of flatness (zero curvature). Such implied additional property of flatness is probably a special case; that is most manifolds probably have additional property of non-zero curvature. Hence even a limited use of the word Euclidean in a definition of manifold, would seem highly misleading, to other than mathematicians of course. http://en.wikipedia.org/wiki/Manifold#Differentiable_manifolds"
 
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  • #4
zankaon said:
most manifolds are non-smooth;that is, non-differential. Hence one could not consider the neighborhood of a 'point' nor a patch of manifold.
:confused: The former has absolutely nothing to do with the latter.

The definition of "topological space" says that every point has a neighborhood. The definition of "manifold" says that every point has a neighborhood with a specific property.

So the concept of tangent space would not even be realizable.
Putting a tangent bundle on a manifold is equivalent to putting a differential structure on the manifold, so those bits are equivalent. But this has nothing to do with neighborhoods or patches.

most manifolds probably have additional property of non-zero curvature.
Manifolds don't have curvature, be it zero or otherwise. Curvature is a property of Riemannian manifolds, and other similar structures.
 
  • #5
To the OP: are you aware that a hemisphere and a disk are homeomorphic? That a square and a circle are homeomorphic? Not agreeing with your intuition <> deficient.

What you seem to be looking for is a local isometry condition: but by Cartan's theorem, this would imply that every manifold has euclidean space as a universal cover. I think you would agree this is a bit restrictive.
 
  • #6
zankaon said:
Conventional manifold definition refers to the neighbor of every point having a Euclidean space description. http://en.wikipedia.org/wiki/Manifold" [Broken] But if most manifolds have additional property of some curvature, then won't such manifold definition actually be describing a tangent space i.e. not part of curved manifold?

The answer to your question is yes. In order to have curvature one needs a tangent space and that requires more than the space being just locally Euclidean. It requires that the coordinate transformations on overlapping charts be at least twice differentiable(I think) since curvature is defined in terms of second derivatives.

I guess there is a question of the degree of differentiability of the coordinate transformations. If they are for instance, only C1, are they compatible with a C2 structure? I think Whitney did the research on this. But C2 should be all you need for curvature.
 
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What is a manifold?

A manifold is a mathematical concept that refers to a topological space that is locally similar to Euclidean space. It can be described as a space that is smooth and has no edges or corners.

How is a manifold defined?

There are several ways to define a manifold, but the most common definition is through charts and atlases. A chart is a map that transforms a piece of the manifold into Euclidean space, and an atlas is a collection of charts that cover the entire manifold.

What is a manifold definition deficiency?

A manifold definition deficiency refers to a situation where a particular definition of a manifold is not sufficient to describe all possible manifolds. This can happen when the definition is too restrictive or does not capture all the necessary properties of a manifold.

Why is it important to address manifold definition deficiencies?

Manifold definition deficiencies are important to address because they can limit our understanding and application of manifolds in various fields, such as physics, engineering, and computer science. By identifying and addressing these deficiencies, we can have a more comprehensive understanding of manifolds and their properties.

How can manifold definition deficiencies be resolved?

Manifold definition deficiencies can be resolved through the development of new definitions or alternative approaches to describing manifolds. This can involve incorporating new concepts or properties that were not previously considered in the definition. It may also require collaboration and discussion among mathematicians and scientists to reach a consensus on a more comprehensive definition.

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