Results 1 to 5 of 5
Like Tree1Thanks
  • 1 Post By xxp9

Math Help - Time derivative of the Levi-Civita Connection (Ricci Flow)

  1. #1
    Newbie
    Joined
    Sep 2011
    Posts
    22

    Time derivative of the Levi-Civita Connection (Ricci Flow)

    Hello,

    I am learning about the Ricci Flow, reading: Lectures on the Ricci flow - Peter Topping (download pdf).

    In Proposition 2.3.1 (i.e. page 28) the expression \frac{\partial}{\partial t}{\nabla}_X Y appears (To understand the equation, you may have to read the few lines above Proposition 2.3.1). I don't know how this is defined. Can someone tell me?

    In addition, I don't know the definition of \langle h,\alpha \rangle in Proposition 2.3.6.

    Regards,
    engmaths
    Last edited by engmaths; April 27th 2013 at 11:49 PM.
    Follow Math Help Forum on Facebook and Google+

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

    Re: Time derivative of the Levi-Civita Connection (Ricci Flow)

    In each time t, let V=DxY is a vector field on M.
    At time 0, d/dt( V ) = (Vt - V0)/t, this can be defined on each point of M.
    <h, a> is the contraction of tensors. The result is a scalar. Just like the contraction between a vector and a co-vector:
    <gradf, v>=<df, v> = v(f) = the directional derivative of f on the direction of v.
    Thanks from engmaths
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2011
    Posts
    22

    Re: Time derivative of the Levi-Civita Connection (Ricci Flow)

    Hello xxp9, that helped me, thank you!

    I got another quick question that I just want to add here:


    Let \alpha be a tensorfield , X a vector field, \varphi a diffeomorphism of a Riemannian Manifold M and L the Lie derivative.

    I want to show:
    \varphi^*(L_X\alpha)=L_{(\varphi^{-1})_*X}(\varphi^*\alpha)

    Let b_t be the local flow of X around p. Then \varphi^{-1}\circ b_t is the local flow of (\varphi^{-1})_*X around p. Then
    L_{(\varphi^{-1})_*X}(\varphi^*\alpha)=d/dt t=0 ((\varphi^{-1}\circ b_t)^*(\varphi^*\alpha))_p = d/dt t=0  ((b_t)^*\alpha)_p=(L_X\alpha)_p
    what seems to be wrong. What is wrong?

    Thanks in advance!
    Last edited by engmaths; May 4th 2013 at 11:32 AM.
    Follow Math Help Forum on Facebook and Google+

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

    Re: Time derivative of the Levi-Civita Connection (Ricci Flow)

    the flow that induces (\varphi^{-1})_* X is \varphi^{-1} \circ b_t \circ \varphi.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Sep 2011
    Posts
    22

    Re: Time derivative of the Levi-Civita Connection (Ricci Flow)

    Thanks xxp9, I looked up the definition of (\varphi^{-1})_* X and get the right solution now.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Question involving Levi-Civita symbol
    Posted in the Advanced Applied Math Forum
    Replies: 4
    Last Post: April 3rd 2013, 04:06 AM
  2. Replies: 5
    Last Post: December 16th 2012, 08:17 AM
  3. Levi-Civita Alternating Symbol
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 29th 2012, 10:58 AM
  4. Levi-Civita Component Count
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: July 19th 2008, 09:00 AM
  5. Time wasting Line Connection Problem
    Posted in the Math Challenge Problems Forum
    Replies: 2
    Last Post: December 6th 2007, 02:24 PM

Search Tags


/mathhelpforum @mathhelpforum