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