Examples of manifolds not being groups

  • B
  • Thread starter trees and plants
  • Start date
  • #1
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.
 

Answers and Replies

  • #2
What if the manifold is not smooth? Another question i have is can a topological space not be a group?
 
  • #3
fresh_42
Mentor
Insights Author
2021 Award
17,591
18,099
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.
 
  • Informative
Likes Klystron
  • #4
martinbn
Science Advisor
2,973
1,303
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.
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
fresh_42
Mentor
Insights Author
2021 Award
17,591
18,099
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.
 
  • Like
Likes Klystron
  • #7
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
fresh_42
Mentor
Insights Author
2021 Award
17,591
18,099
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.
 

Suggested for: Examples of manifolds not being groups

Replies
4
Views
177
Replies
2
Views
304
  • Last Post
Replies
1
Views
1K
Replies
10
Views
1K
Replies
1
Views
717
Replies
2
Views
542
Replies
4
Views
791
Replies
6
Views
1K
  • Last Post
Replies
2
Views
599
  • Last Post
Replies
7
Views
629
Top