Applications of topological spaces not homeomorphic to R^n in physics

[trees and plants]

Hello. So, the question is do you know any applications of topological spaces which are not homeomorphic to R^n in physics? Motivation for the question i am making: as i think if a topological space is homeomorphic to R^n then differential calculus is allowed on it. Modern physics uses i think differential equations so i am interested in learning if these kind of topological spaces are used in physics and physicists have considered them interesting for applications in physics. Thank you.

 

[fresh_42]

That is exactly how Riemannian manifolds came into play and why they are considered. Other examples in physics are Hilbert spaces, Sobolev spaces etc.

 

[trees and plants]

They are homeomorphic near each point with R^n but globally they could be not homeomorphic to R^n, am i correct? Homeomorphism with R^n near each point is used for differentiable manifolds? Is there any other way to define differentiability in a topological space? Perhaps not? Or an equivalent of this? Or a more general?

 

[fresh_42]

They are homeomorphic near each point with R^n but globally they could be not homeomorphic to R^n, am i correct?

Yes.

Homeomorphism with R^n near each point is used for differentiable manifolds? Is there any other way to define differentiability in a topological space?

Not that I am aware of. Differentiability is a local phenomenon. It is a linear approximation of a otherwise curved space. Hence you need a linear structure where the derivatives live in, i.e. a Euclidean space. This means that local flatness is necessary to transport the concept into known areas. There are several generalizations which are not directly related to differentiable manifolds. E.g. the boundary operator in homological structures, or derivations of algebras can be considered as a form of derivatives; or different concepts like the Schwarzian derivative.

 

[trees and plants]

Yes.
Not that I am aware of. Differentiability is a local phenomenon. It is a linear approximation of a otherwise curved space. Hence you need a linear structure where the derivatives live in, i.e. a Euclidean space. This means that local flatness is necessary to transport the concept into known areas. There are several generalizations which are not directly related to differentiable manifolds. E.g. the boundary operator in homological structures, or derivations of algebras can be considered as a form of derivatives; or different concepts like the Schwarzian derivative.

You mean a vector space, with linear maps?Having the properties of linear maps? Is this enough for differential calculus to be allowed on a set? Or more conditions are needed?

 

[fresh_42]

You mean a vector space, with linear maps?Having the properties of linear maps? Is this enough for differential calculus to be allowed on a set? Or more conditions are needed?

You also need a method to measure the accuracy of the linear approximation, i.e. something that makes it an approximation, not just an arbitrary linear space.

 

[trees and plants]

You also need a method to measure the accuracy of the linear approximation, i.e. something that makes it an approximation, not just an arbitrary linear space.

Bus as i read for differentiable manifolds someone needs transition maps, not only a homeomorphism for the neighborhood to be homeomorphic to R^n at each point. Is this correct?

 

[fresh_42]

These are two different procedures. Differentiation on a manifold means, we take (and therefore need) a local chart, which is a flat map of the location we are at, transport the problem onto the chart, differentiate there as usual, since it is a copy of some $\mathbb{R}^n$, and "upload" the result from the map into the manifold.

You can do this with any roadmap. Look out for a bended road you know and you have a map from. Then ask: If I would drive too fast on that road, where in the wild would I end up? Then take your map, search for the point where you are too fast and draw the tangent to it. That gives you the direction along which you will fly into the bushes. Go back to the road and see where the tangent points to in real life. This procedure is essentially what's going on if we differentiate on a manifold (real environment) using a chart (roadmap).

The other topic which we were currently talking about was the differentiation itself. We have that bended road on the map and want to draw a tangent, i.e. we are already in the $\mathbb{R}^n$. I like the Weierstraß notation, for it is short and shows what happens:
$$\mathbf{f(x_{0}+v)=f(x_{0})+J(v)+r(v)}$$Equation (1) in https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/
$\mathbf{J}$ is the (linear) derivative, $\mathbf{v}$ the direction of the tangent, $\mathbf{r}$ the error margin of the approximation, and $\mathbf{x_0}$ the location where all this takes place. $\mathbf{f}$ is the road on the map.

 

[trees and plants]

Another question i have is in terms of a space allowing differentiability on it, what are the most general kind of geometric spaces so far that do this? Are they differentiable manifolds? And another one, when a vector space is differentiable? Should it at least be a manifold?

 

[fresh_42]

If we restrict ourselves to the classical concept, and not allow the other more general views as in the examples above, I'd say yes: we need a differentiable manifold. Look at Weierstraß' formula: we need something which gives meaning to the quantities it uses.

 

[mathwonk]

As a slight generalization, one can also discuss differentiability for a space S that is given as an embedded subspace of a differentiable manifold M. Then a map into S is differentiable if it is so as a map into M, and a map out of S is differentiable if it is locally the restriction of a differentiable map out of M. In particular if S is itself a manifold embedded in R^n, this technique allows one to discuss differentiation for S without using local charts. This is often done in elementary treatments to simplify the presentation. e.g. Milnor's Topology from the differentiable viewpoint. But it can also be applied to spaces that are not manifolds, but are embedded in manifolds like R^n. E.g. one can discuss differentiability this way for a figure eight, or a cusp in the plane. A related technique, but more algebraic, is also used in algebraic geometry to discuss differentiability for spaces defined by algebraic equations, but which may not be everywhere manifolds, i.e. for "singular algebraic spaces". These may be embedded in affine space, projective space, or even may be abstract spaces, but equipped with a "structure sheaf" of functions which are differentiable essentially by fiat. For this one may consult Serre's "Faisceaux algebrique coherent".

https://www.jstor.org/stable/1969915?origin=crossref&seq=1

If you prefer English to French, here is a link to a translation:

https://mathoverflow.net/questions/14404/serres-fac-in-englishActually Serre is much too abstract and advanced for a naive intro, if interested, try Undergraduate algebraic geometry, by Miles Reid.

 

[WWGD]

If we restrict ourselves to the classical concept, and not allow the other more general views as in the examples above, I'd say yes: we need a differentiable manifold. Look at Weierstraß' formula: we need something which gives meaning to the quantities it uses.

How about Frechet spaces and the like?

 

[WWGD]

Spaces unlike R^n that I believe are used in Physics ( Don't ask for details) are the L^p spaces and other function spaces you define Fourier series in $L^2[a,b]$. For one, these are infinite-dimensional, so not like R^n.

 

[fresh_42]

How about Frechet spaces and the like?

Fréchet differentiability is not much different than the Weierstraß formula, just that the vectors and function live in Banach spaces, and the linear approximation is an operator.

 

[martinbn]

Hello. So, the question is do you know any applications of topological spaces which are not homeomorphic to R^n in physics?

Anything compact will not be homeomorphic to $\mathbb R^n$. For example spheres.

Motivation for the question i am making: as i think if a topological space is homeomorphic to R^n then differential calculus is allowed on it. Modern physics uses i think differential equations so i am interested in learning if these kind of topological spaces are used in physics and physicists have considered them interesting for applications in physics. Thank you.

Differential calculus is local, you only need things to be locally like $\mathbb R^n$, so any smooth manifold will do.