What is the Role of Lie Groups in Isometry Actions on Spaces?

In summary, the group G acts by isometries on a space X, and the language used is not clear to the speaker. The speaker is unsure if they need to show that the action preserves distance, and asks for clarification. Another person mentions that if X is a Riemannian manifold, the differential of the action should preserve the metric tensor. The speaker then mentions that they are only told about a linear group acting on S^3 and there is a question about geodesics, suggesting that it may indeed be a Riemannian manifold.
  • #1
WWGD
Science Advisor
Gold Member
7,347
11,314
Hi, everyone:

I am asked to show that a group G acts by isometries on a space X.

I am not clear about the languange, does anyone know what this means?.

Do I need to show that the action preserves distance, i.e, that

d(x,y)=d(gx,gy)?.

Thanks.
 
Physics news on Phys.org
  • #2
Depends on context, but if it is a Riemannian manifold, presumably you want to show that the differential of the action preserves the metric tensor.
 
  • #3
Thanks. I am only told of a linear group, i.e, a group of matrices
acting on S^3. There is a question on geodesics, so you may
be right, and we may need to consider this as a Riemannian mfld.
 

FAQ: What is the Role of Lie Groups in Isometry Actions on Spaces?

1. What are Lie groups?

Lie groups are mathematical objects that are used to study continuous symmetry. They are groups that have both a smooth manifold structure and a group structure, making them useful in many areas of mathematics and physics.

2. How are Lie groups different from other groups?

Lie groups are different from other groups because they have a smooth manifold structure, which means they are continuously differentiable and can be described by smooth functions. This allows for more advanced mathematical analysis and applications in physics.

3. What are some real-world applications of Lie groups?

Lie groups have many real-world applications, particularly in physics and engineering. They are used to study the symmetries of physical systems, such as in quantum mechanics and relativity. They are also used in robotics and computer graphics for motion planning and control.

4. What is the relationship between Lie groups and Lie algebras?

Lie groups and Lie algebras are closely related mathematical objects. Every Lie group has an associated Lie algebra, which is a vector space that captures the infinitesimal information about the group's structure. In turn, Lie algebras can be used to classify and study Lie groups.

5. Are there different types of Lie groups?

Yes, there are different types of Lie groups, including compact and non-compact Lie groups, simple and non-simple Lie groups, and connected and non-connected Lie groups. These types have different properties and applications, and they are classified based on their structure and symmetries.

Similar threads

A Differential structure of the group of automorphism of a Lie group
Replies
0
Views
883
A What Is a Lie Group?
Replies
11
Views
3K
I Derivative of the Ad map on a Lie group
Replies
3
Views
2K
Isometry group of positive definite matrices
Replies
1
Views
1K
A About computing the tangent space at 1 of certain lie groups
Replies
4
Views
2K
A In what representation do Dirac adjoint spinors lie?
Replies
1
Views
2K
A Compact Lie Group: Proof of Discrete Center & Finite Size?
Replies
14
Views
2K
About lie algebras, vector fields and derivations
Replies
20
Views
2K
A Derivative of smooth paths in Lie groups
Replies
10
Views
2K
A Characterizing the adjoint representation
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top