Trying to get my head around tangent bundles

[mikeph]

Hello,

Say you have a function f on the domain R^n, and an integral transform P which integrates f over all possible straight lines in R^n. I am lead to believe that the range of this is R^(2n), or a tangent bundle, which I am having MASSIVE problems visualising!

Am I right in saying the tangent bundle can be described by the multiplication of a vector on the unit sphere in R^n with a vector in R^n, ie. all points, then from each point, subtending all angles?

But surely this creates duplication? ie. for n=3, the line passing point (0,0,0) parallel to (1,0,0) must be the same as the line passing through (1,0,0) parallel to (1,0,0). So I am trying to picture a more "efficient" way to specify the range of this transform...

Is it completely described by all vectors in R^n perpendicular to each plane described by the vector on the unit sphere in R^n? How many are there per plane?

SO confused! But intrigued...

Thanks,
Mike

 

[OrderOfThings]

The set of oriented lines in Rⁿ is isomorphic to the tangent bundle of the n-1-sphere TS^{n-1}. At each point on a tangent plane of the unit sphere there is exactly one line intersecting orthogonally at that point. Choose one orientation of the line, say outwards, and this gives the isomorphism.

 

[mikeph]

Thankyou,

Can I confirm, does isomorphism mean all the lines parallel to that line?

I would think the space which this transform maps to is R^(n-1)^2...

The reason being, each hyperplane in R^n can be described as perpendicular to a point on the unit sphere in R^n, requiring n-1 scalars. And then for each plane you can completely parametrise the perpendicular lines crossing it using R^(n-1), since the plane itself is parametrised this way.

So pick a direction, then pick a point on the plane perpendicular to this, and you get a unique straight line through R^n.

Is that correct?

Thanks,