straight lines on closed manifolds

Higher-dimensional geometry (previously "Polyshapes").

straight lines on closed manifolds

Postby bo198214 » Thu Aug 10, 2006 11:19 am

I found a very interesting question on another forum:
If one starts going always in the same direction does one then return to the starting point (in our universe)?

I mean we dont know so much about the universe and a recent theory proposes the universe being a dodecahedron with opposite faces identified. So I generalize the question a bit:
Are there straight lines on closed manifolds that are not closed?

But indeed this question can be answered with 'yes', as is hopefully clear from the following pictures (without further explanation). We take as counterexample the 2dim torus:

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Perhaps it should be added that the 2-torus is submanifold of the 3-torus and so if our universe would be a 3-torus there would be directions to start with, such that we would never return to our starting point.

What now really is not obvious to me, whether there are closed manifolds, with points, where *every* straight line through this point is not closed. I.e. regardless what initial direction we start our walk, we would never return to our starting point.

Another associated question is: If we start in a direction that does not return to the starting point on a closed manifold, do we then automaticly/generally come arbitrary close to our starting point (as is the case with the 2-torus example).
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Postby bo198214 » Mon Aug 14, 2006 11:34 am

Hey guys (n girls (at Wendy :) )), I could inquire an answer about this from sci.math.research. Every closed manifold has for every point a closed geodesic (i.e. straight) line going through that point. See the complete answer here.
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Postby wendy » Tue Aug 15, 2006 7:03 am

I did look at some rather bizarre spaces, but every one of them had some sort of set of finite paths. Many of these i don't know the names for, but the cells were of these.

Note that not every tiling-cell serves as a torus, it must meet further conditions also.

x5o3o3o Poincare Dodecahedron
x5o3o5o Dodecahedron
m3o5oAo Rhombo-dodecahedron
x3o5o3x Icosahedron
x4o3o4o Cube
m3o:s3s4s Gyrated cube. (this is weird)
m3o:s3s5s Gyrated dodecahedron
o3x4x3o truncated cube

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