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I see the Alexandrov theorem wich say that every comapct, without boundary embedded in Euclidien space must be a round sphere. Can someone explain for me the difference between geodesic sphere rond sphere and hypersphere? Thank you.

Mohammed Mohammed
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  • I thought "hypersphere" just meant going in higher dimensions, i.e. the n-dimensional analogue of a sphere, similar to (hyper)plane or (hyper)cube. – TMM Apr 23 '17 at 20:15
  • Thank you fr the answer, the doublt I ave is also about the geodesic sphere and the difference with the sphere – Mohammed Mohammed Apr 23 '17 at 20:37
  • "every comapct, without boundary embedded in Euclidien space must be a round sphere" Can you clarify this sentence? –  Apr 23 '17 at 22:55
  • Sorry for the message. Th Alexandrov theorem is:Let M be a compact hypersurface embedded in the Euclidean space Rn+1. with constant mean curvature. Then M is a round sphere. – Mohammed Mohammed Apr 24 '17 at 16:01

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$\textbf{Update}$: In response to what I wrote below, since there isn't literature I could find on the "geodesic sphere", I choose to go with what seemed most natural. Why this definition is natural for me, you can read in the comments below.

The hypersphere is just the $n$-dimensional analogue for an $n$-sphere when $n>3$. The geodesic sphere is the sphere made from geodesics. This can be different from the standard sphere, which can be a smooth, topological, etc type of embedding i.e there doesn't have to be an isotopy between these other embeddings and the geodesic sphere.

Faraad Armwood
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  • I've never heard of a "geodesic sphere," other than the popularized "geodesic dome." – Ted Shifrin Apr 23 '17 at 20:49
  • @TedShifrin When I read the question I figured, what else could it mean? and what is the most logical definition for it? The correspondence with the geodesic dome is that it is a linear approximation of the smooth sphere i.e on the geodesic dome, the geodesics are straight lines in the literal sense, due to the fact that it is a polyhedron. And so I guess one could say that the dome is an example of the geodesic sphere when you use a flat metic. – Faraad Armwood Apr 23 '17 at 20:52