# 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.