Results 1 to 2 of 2

Math Help - Differential geometry question

  1. #1
    Member
    Joined
    Mar 2010
    Posts
    122

    Differential geometry question

    Let p be a point on a curve γ on a surface S, and let Π be the tangent plane to S at p. Let μ be the curve obtained by projecting γ orthogonally onto Π . Show that the curvature of the plane curve μ at p is equal up to sign,to the geodesic curvature of γ at p.

    I have no idea how to prove this. All i know than is that the normal curvature is given by Lu'^2 +2Mu'v' +Nv'^2
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23
    Let T be the tangent vector( velocity) of γ, then dT/ds is the curvature vector( acceleration), and the norm of its projection onto Π is the geodesic curvature. Then the statement is a consequence of the following statement:
    Projection of the acceleration of a curve equals to the acceleration of its projection.
    This is easy to see if your choose a coordinate system so that the projection is (x,y,z) |->(x,y), so the acceleration is (x'', y'', z''), its projection is (x'',y''). And the projection of the curve is (x,y), its acceleration is (x'',y''). DONE.
    The deep idea is, the projection onto Π is an isometry in a infinisimal neighborhood of p.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Differential Geometry - Question about Limit
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: November 18th 2011, 04:52 AM
  2. do Carmo question - differential geometry....
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 5th 2011, 01:51 PM
  3. Differential geometry question
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 1st 2010, 04:07 AM
  4. Differential Geometry Bias Question
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: July 6th 2010, 04:37 PM
  5. Differential geometry question
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: December 7th 2007, 07:50 AM

Search Tags


/mathhelpforum @mathhelpforum