Results 1 to 4 of 4

Math Help - compact surfaces question

  1. #1
    Newbie
    Joined
    Dec 2012
    From
    Los Angeles
    Posts
    12

    compact 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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23

    Re: 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)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2013
    From
    Los Angeles
    Posts
    2

    Re: compact surfaces question

    this doesnt make sense can you explain part a more clearly?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23

    Re: 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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. compact subsurfaces of bordered surfaces of infinite genus
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: February 17th 2013, 12:17 AM
  2. Question on Quadric Surfaces
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 20th 2012, 09:39 PM
  3. Replies: 1
    Last Post: November 19th 2011, 06:32 AM
  4. Question about the triangulation of surfaces
    Posted in the Geometry Forum
    Replies: 0
    Last Post: March 5th 2011, 12:13 AM
  5. A question on Riemann surfaces and spaces of germs
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: November 17th 2010, 08:11 PM

Search Tags


/mathhelpforum @mathhelpforum