Results 1 to 2 of 2

Math Help - Tangent bundle is always orientable

  1. #1
    Newbie
    Joined
    Aug 2009
    Posts
    11

    Tangent bundle is always orientable

    Hello all, trying to prove that the tangent bundle TM of a smooth n-manifold M is always orientable. To make it easier to write down Im doing n=2.

    Let the differentiable structure on M be given by \{U_\alpha,\phi_\alpha\}. This then induces a differentiable structure on TM given by \{U_\alpha\times\mathbb{R}^2,\psi_\alpha\} where we define \psi_\alpha:U_\alpha\times \mathbb{R}^2\to TM,\;\; (u_\alpha,v_\alpha,x,y)\mapsto \left(\phi(u_\alpha,v_\alpha),x\frac{\partial}{\pa  rtial u_\alpha}+y\frac{\partial}{\partial v_\alpha}\right) i.e. we take as the coordinates of a point (p,w) in TM the coordinates of the point p in U, plus the coordinates of w in the basis \{\partial/\partial u_\alpha,\partial/\partial v_\alpha\}.

    Once you've proved that TM is a smooth 4-manifold (which isn't hard) you then have to show that for every change of coordinate function the determinant of its jacobian matrix is positive, i.e.
    \det(d(\psi_\beta^{-1}\circ\psi_\alpha))>0 for all \alpha,\beta.

    To write out the 4x4 matrix d(\psi^{-1}_\beta\circ\psi_\alpha) is where I've come unstuck. The first two columns will be the partial derivatives of the 4 component functions of \psi^{-1}_\beta\circ\psi_\alpha with respect to u_\alpha,v_\alpha, i.e. the top-left 2x2 block will just be d(\phi_\beta^{-1}\circ\phi_\alpha) and the bottom-left will be zero because d^2=0.

    I don't know how to work out the two 2x2 blocks on the right hand side of the matrix. Basically I don't know what to take the partial derivative with respect to, and just how to write it down. Hopefully the top-right will be zero and the bottom-right will be d(\phi_\beta^{-1}\circ\phi_\alpha) as well, so then we have as a determinant the square of a real number which will always be positive.

    Any help anyone could give me would be great!

    Cheers,

    Atticus
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Aug 2009
    Posts
    11
    Giving this a bump in case anyone has any ideas. Getting some help on it today though.

    Atticus
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Tangent Bundle of S^1
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: November 14th 2011, 06:22 AM
  2. Tensor bundle
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: February 7th 2011, 07:50 AM
  3. Tangent bundle of a Lie Group is trivial
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: January 17th 2011, 06:06 PM
  4. [SOLVED] Tangent bundle of submanifold
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: December 4th 2010, 06:44 PM
  5. Lie group and the frame bundle
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 20th 2010, 11:42 AM

Search Tags


/mathhelpforum @mathhelpforum