Can a sphere have flat geometry?

[Hoku]

I know that the vocabulary for topology/geometry is complex. Please bear with me if I'm not using the appropriate terminology. I'm trying to get a grasp on my question as much as I am trying to find an answer to it.

Is it possible for something to disperse throughout a spherical space if the basic stucture of that space is flat? I mean flat in 3-dimensions, like a lattice structure that is flat instead of curved.

I'll try rephrasing the question:

If the shape of space is spherical, does it matter whether or not it's geometry is flat or curved? If so, what are the consequences of each scenario?

 

[Ben Niehoff]

A topological sphere can have a geometry that is flat except at a finite number of points: for example, the regular polyhedra are spheres that are flat except at their vertices. At the vertices themselves, the curvature becomes infinite, in such a way that its integral gives the deficit angle at that vertex.

There are "index theorems" which relate the total curvature integrated over a manifold to that manifold's topological invariants. In particular, for a 2-manifold, the integral of the Ricci scalar gives (2 pi times) the Euler characteristic. The Euler characteristic of a sphere is 2, so there must be some curvature somewhere if a manifold is to be topologically a sphere.

 

[Hoku]

Ben, thank you for your insights! As I said, I'm also trying to understand my question here, so I'd like to begin refining it...

What if you're dealing with a 3-dimensional sphere that lacks a 2-dimensional "skin"? Such a thing exists, doesn't it? Wouldn't it somehow be similar to a 3-torus if it lacked the skin?

 

[Studiot]

If you mean what I think you mean by 'skin' you need to be careful of terminology here.

Technically a sphere is a surface.

Check your definitions of

Sphere
Open Ball
Closed Ball

http://en.wikipedia.org/wiki/Ball_(mathematics )

 

[Hoku]

Thanks for clearing that up, Studiot. That's why I gave warning that I might not have the right terminology. I did look at the wiki link but the math is over my head. Is the math describing an internal geometry? If an open ball even has geometry, can it have FLAT geometry?

I'm trying to apply this to an absolute spacetime - spacetime that the big bang happened in instead of spacetime that the big bang created. People say that spacetime is flat except where mass curves it. I'm trying to understand, on a simple level, if that means spacetime must be infinite in all directions or if flat space with points of curvature can also be an open ball.

As Ben pointed out, A sphere must have at least some curvature somewhere but I think a ball can be described with flat geometry. Am I right?

 

[mma]

Since we don't know the topology of the spacetime, we always talk only about a part of it. If this part is homeomorphic with a 4-dimensional ball, then of course you can endow it by flat metric if you want. But in physics metric is not arbitrary because it is used to describe gravitation. Perhaps gravitation can also be described by a metric having torsion instead of curvature (see http://en.wikipedia.org/wiki/Teleparallelism" ). But "flat" means curvature-free and torsion-free. So I'm afraid that a flat metric of spacetime can't have any physical significance. However, gravitational-like phenomena can occur in flat geometry too, for example in an accelerating elevator, being far from any masses, the spacetime is flat. Inertial forces are due to the nonvanishing components of the Christoffer-symbols, rather than curvature.