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Thanks for the reply. I am new to differential Geometry. I have quoted this post because it is a good example to further explain my question. Now, if you run a straight line from NP to SP that is unique. these other Longitude lines that run on the surface are not unique. So my question is (I guess) can we have a unique STRAIGHT line in higher than 3D.

also, I have read that the concept of line(I guess they mean straight line) in higher than 3D does not make sense or something to that effect.

If my question is not making sense please tell me.

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In curved spaces it may not be true for all points (the case already given of antipodal points on a sphere is an example). And in some spaces, the notion of "straight line" is not meaningful, for instance a manifold without a defined metric. But these considerations don't have anything to do with dimension.

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Somehow, I don't think you're meaning to say "straight line", but rather "geodesic". A geodesic is a shortest line connecting two points (this isn't the definition of a geodesic, but I'm being informal). Geodesics make sense in a very general context. The concept of "straight line" only really makes sense in a vector space (this is basically just ##\mathbb{R}^n##, so ##n##-dimensional flat space).Thanks for the reply. I am new to differential Geometry. I have quoted this post because it is a good example to further explain my question. Now, if you run a straight line from NP to SP that is unique. these other Longitude lines that run on the surface are not unique. So my question is (I guess) can we have a unique STRAIGHT line in higher than 3D.

also, I have read that the concept of line(I guess they mean straight line) in higher than 3D does not make sense or something to that effect.

If my question is not making sense please tell me.

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s^2=dx1^2 +dx2^2 +dx3^2 +dx4^2 + ....

for 3D dx=1 line element length =sqrt(3)

for 4D dx=1 line element length =sqrt(4)

for 3D that is clear but for 4D do I actually walk that much if I go into that space, sounds very strange.

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WannabeNewton

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Hi Tiny! If by "straight line" you mean geodesics then you can in fact define geodesics without any riemannian metric as long as you have an affine connection ##\nabla## endowed on yourAnd in some spaces, the notion of "straight line" is not meaningful, for instance a manifold without a defined metric.

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Do you mean straight line or geodesic [as the previous posters pointed out?] Assuming you really do mean straight line, then I am still not exactly sure what you mean: let's back down a bit, how about line in 1D [just line by itself], line in 2D [line in a plane], and line in 3D? You see that any line in 3D space can really be contained in a plane, so there is really no difference between a line in 2D or 3D or 4D -- they are one-dimensional entity, what exactly is confusing you?

s^2=dx1^2 +dx2^2 +dx3^2 +dx4^2 + ....

for 3D dx=1 line element length =sqrt(3)

for 4D dx=1 line element length =sqrt(4)

for 3D that is clear but for 4D do I actually walk that much if I go into that space, sounds very strange.

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yes, I meant a straight line and not a geodesic. Of course I can understand it in the sense of the mathematics, well sort of. But when in 4D and higher you cannot see it as a line (or anything else for that matter) in the usual "physical" representation, so I don't know really what to make of it.Do you mean straight line or geodesic [as the previous posters pointed out?] Assuming you really do mean straight line, then I am still not exactly sure what you mean: let's back down a bit, how about line in 1D [just line by itself], line in 2D [line in a plane], and line in 3D? You see that any line in 3D space can really be contained in a plane, so there is really no difference between a line in 2D or 3D or 4D -- they are one-dimensional entity, what exactly is confusing you?

So, in that sense whether it is GR or string it seems what they imply is that our reality is indeed a mathematical entity, at least this my naive interpretation.

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