Problem Set Three

Problem 1

Find coordinates for the manifold M consisting of pairs: points in three space and planes passing through the point. This will be a five dimensional manifold, since there are two degree of freedom in the orientation of the plane. We can describe the plane with homogenous coordinates [k,l,m], this being the plane parallel to the plane through the origin
k x + l y + m z = 0
Thus we could describe all of the manifold with the pair: {(x,y,z),[k,l,m]}, and if you want coordinate charts, you can use the usual trick to generate charts like
(x, y, z, k/m, l/m) provided m nonzero
and so on.

Problem 2

Describe the map from the surface of the sphere into three space in usual position (I sort of forgot to include this in the problem statement), with the associated tangent plane.
We can describe the sphere by homogeneous coordinates restricted to equivalence under positive factors: [x,y,z]. The map will then be, using the above coordinates
[x,y,z] \mapsto
( x/Sqrt(x^2+y^2+z^2), y/Sqrt(x^2+y^2+z^2), z/Sqrt(x^2+y^2+z^2), [x,y,z])
This everywhere respects the homogeneous of the original description, and so it is well defined. To see that the tangent plane at (x,y,z) is given by [x,y,z], just look at the gradient of the function x^2 + y^2 + z^2 , which is a 1-form whose zero surface is the tangent plane.