Let M be a closed, orientable, and bounded surface in R3
a) Prove that the Gauss map on M is surjective
b) Let K_{+}(p)= max{0, K(p)} Show that the integral over the surface M, ∫ K_{+} dA ≥ 4π. Do not use the Gauss Bonnet theorem to prove this
Let M be a closed, orientable, and bounded surface in R3
a) Prove that the Gauss map on M is surjective
b) Let K_{+}(p)= max{0, K(p)} Show that the integral over the surface M, ∫ K_{+} dA ≥ 4π. Do not use the Gauss Bonnet theorem to prove this
a) For any unit vector v, there exists a plane P with v as its normal vector, and has no intersection with M( since M is bounded).
P defines a continuous function on M, d: M -> R, d(p) is the distance between p and P. Since M is compact, d has a minimum. That is, there is a point p on M that is closes to P.
The unit normal vector of M on p will be v. This proves that the Gauss map is surjective.
b) ∫ K+ dA >= ∫ K dA = ∫ da = 4π. Where the last integral ∫da is performed on the unit sphere( since the Gauss map is surjective its image covers the whole sphere)
To prove that the Gauss map G is surjective, we need to show that for every point v on the unit sphere S, there is a point p on M, so that G(p)=v.
To find p, fix a plane which is orthogonal to v and far away from M, then p is the point on M that is closest to the plane.