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Quick question: from which topological consideration can one derive the fact that a sphere S² does not allow for a globally flat geometry?
quasar987 said:Because it would mean that there is a diffeomorphism S^2 -->R^2
tom.stoer said:Quick question: from which topological consideration can one derive the fact that a sphere S² does not allow for a globally flat geometry?
quasar987 said:Ok, I thought globally flat meant there exists a global isometry from S^2 + some metric to R^2 + usual metric.
tom.stoer said:@lavinia: thanks for reminding me to these facts
Let me see if I understand. The Euler characteristic of S² is 2. This can be seen by using a triangulation on S² or by calculating
[tex]\chi_{S^2} = 2 - 2g_{S^2}[/tex]
and using
[tex]g_{S^2} = 0[/tex]
But by Gauss-Bonnet we know that
[tex]\chi_{S^2} = \frac{1}{2\pi}\int_{S^2}K[/tex]
Global flatness would imply [tex]K = 0[/tex] globally which results in [tex]\chi_{S^2} = 0[/tex] which is a contradiction.
arkajad said:Well, an infinite cyllinder has a globally flat geometry, but there is no diffeomorphism from the cylinder onto R^2.
But you may check this ion Wikipedia: "Vector fields on spheres". Global flatness of S^2 would imply existence of two nowhere vanishing vector fields.
lavinia said:generally global flatness only implies the existence of one nowhere vanishing vector field.
arkajad said:S^2 is two-dimensional, connected and simply connected. You choose two orthonormal tangent vectors at one point and transport them using parallel transport to every other point. This transport is path independent (zero curvature plus topology). This way you would get two orthonormal vector fields. While by Adams' theorem even one does not exist.
lavinia said:this is wrong. There are no independent vector fields on the 2 sphere.
tom.stoer said:By inspection of 2-2g = 0 only g=1 allows for globally flat geometries.
Another related question: The Gauss-Bonnet theorem restricts the existence of globally flat geometries. What about the other way round? Suppose I have a manifold for which I can proof that the Euler characteristic vanishes. Does this automatically guarantuee that a globally flat geometry must exist? I guess not, but what are the other restrictions?
arkajad said:1) That is what I was saying, assuming hypothetically global flatness - as in the original question. Proof by contradiction. There is not even one.
2) Whether a given manifold is globally flat or not depends on the Riemannian metric. R^2 can be made globally flat and can be made non-flat even locally. Up to you.
tom.stoer said:By inspection of 2-2g = 0 only g=1 allows for globally flat geometries.
Another related question: The Gauss-Bonnet theorem restricts the existence of globally flat geometries. What about the other way round? Suppose I have a manifold for which I can proof that the Euler characteristic vanishes. Does this automatically guarantuee that a globally flat geometry must exist? I guess not, but what are the other restrictions?
"No globally flat geometry on S²" refers to the fact that it is impossible to create a flat map of the Earth's surface without distorting its shape or size. This is because the Earth's surface is a curved two-dimensional space, called a sphere, and therefore cannot be represented accurately on a flat surface.
This is because the Earth is not a flat surface, but rather a curved two-dimensional space. In order to accurately represent the Earth's surface on a map, we would need to use a three-dimensional model or a projection that accounts for the Earth's curvature.
S² refers to the two-dimensional surface of a sphere, which is a common way to represent the Earth's surface. In this context, S² is used to describe the curved surface of the Earth, rather than a flat surface.
Yes, there have been many attempts to create a flat map of the Earth's surface, but all of these maps involve some level of distortion. This is because it is mathematically impossible to accurately represent a curved surface on a flat surface without distorting its shape or size.
This concept of "no globally flat geometry" applies to any curved surface, not just the Earth's surface. It is a fundamental principle in mathematics that a curved surface cannot be accurately represented on a flat surface without distortion.