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

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- May 11th 2013, 11:28 AMhenderson7878compact surfaces question
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 - May 11th 2013, 02:12 PMxxp9Re: compact surfaces question
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) - May 12th 2013, 01:52 PMmath410Re: compact surfaces question
this doesnt make sense can you explain part a more clearly?

- May 12th 2013, 03:38 PMxxp9Re: compact surfaces question
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.