Examples of manifolds not being groups

In summary, the conversation discusses examples of manifolds not being groups, the role of smoothness in defining manifolds, and the requirements for a space to be considered a geometric space. It is also mentioned that manifolds may not necessarily be differentiable. The conversation ends with the suggestion to read about these definitions for further understanding.
  • #1
trees and plants
Hello there. Do you know any examples of manifolds not being groups?Can you talk about some of them developing them as much as you want?Thank you.
 
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  • #2
What if the manifold is not smooth? Another question i have is can a topological space not be a group?
 
  • #3
Analytical groups are an important example, but the group structure isn't necessary for manifolds. E.g. choose a polynomial ##p=p(x_1,\ldots,x_n)## and consider all points with ##p=0##. Then there will be no prior group structure, means: unless you have chosen special polynomials where you can attach a group structure, e.g. ##x_1^2+x_2^2=0##, you won't have one.

Smoothness is pure convenience. You can define manifolds on any ##C^n## level. It is only a requirement for the charts. Take a piece of paper, fold it and you have a continuous manifold which is not differentiable at all.

Topological spaces are rarely a group. Either choose a classical example like a Sierpi´nsky space, Cantor sets, or any other of the usual suspects.

Groups which are manifolds (not the other way around!) have originally been called continuous groups, in contrast to discrete or finite groups. They were studied in calculus of variations. Hence they needed a structure on which calculus can be performed. The least requirement is thus continuity, smoothness the requirement for lazybones who do not want to manage the degree of differentiability at each single step of the considerations. Luckily the major matrix groups are all smooth, as they are defined via polynomials.
 
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  • #4
universe function said:
Hello there. Do you know any examples of manifolds not being groups?Can you talk about some of them developing them as much as you want?Thank you.
universe function said:
What if the manifold is not smooth? Another question i have is can a topological space not be a group?
All of these questions are strange. If you know what the terms you reffer to mean, then the answers should be obvious. If you don't, the the answers are not what you need.
 
  • #5
Obviously one can make sets that are not groups but what about spaces that are not groups?What makes a space a geometric space?
 
  • #6
universe function said:
What makes a space a geometric space?
A metric.

And continuity makes a space a topological space.
And a multiplication makes a space a group.
And differentiability makes a space a manifold.
 
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  • #7
fresh_42 said:
A metric.

And continuity makes a space a topological space.
And a multiplication makes a space a group.
And differentiability makes a space a manifold.
Is a vector space a geometric space?What makes it or not a geometric space?But can a manifold not be a geometric space? I think that manifolds may not be differentiable.
 
  • #8
universe function said:
Is a vector space a geometric space?
No, but often it is. You do not need a metric in a vector space, but many have.
What makes it or not a geometric space?
Still a metric: angles and distances.
But can a manifold not be a geometric space?
Yes, i.e. locally no, since it has homeomorphic charts, globally yes, because you normally don't have only one chart.
I think that manifolds may not be differentiable.
Yes. As I already told you. Continuity is sufficient.

As the questions are answered and we are circling around definitions, this thread is now closed. Please read about those definitions on Wikipedia, nLab, or any other dictionary site.
 

Related to Examples of manifolds not being groups

1. What is a manifold?

A manifold is a mathematical concept that describes a space that looks locally like Euclidean space. In other words, it is a space that is smooth and has no sharp corners or edges.

2. What is a group?

A group is a mathematical structure that consists of a set of elements and a binary operation (such as addition or multiplication) that satisfies certain properties, such as closure, associativity, and the existence of an identity element and inverses.

3. Can a manifold be a group?

No, a manifold cannot be a group. This is because a manifold is a geometric object, while a group is an algebraic structure. Manifolds and groups are two different types of mathematical objects and cannot be directly compared.

4. What are some examples of manifolds that are not groups?

Examples of manifolds that are not groups include the real line, the sphere, and the torus. These are all smooth spaces that do not have the necessary algebraic structure to be considered groups.

5. Are there any manifolds that can be groups?

Yes, there are some manifolds that can be groups, but they are not the typical examples of manifolds. For example, the real line with addition as the binary operation is a group, but it is a one-dimensional manifold. In general, manifolds that are also groups are rare and have specific properties that make them both geometrically and algebraically interesting.

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