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 $\displaystyle \frac{\partial}{\partial t}{\nabla}_X Y$ 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 $\displaystyle \langle h,\alpha \rangle$ in Proposition 2.3.6.

Regards,

engmaths

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.

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

I want to show:

$\displaystyle \varphi^*(L_X\alpha)=L_{(\varphi^{-1})_*X}(\varphi^*\alpha)$

Let b_t be the local flow of X around p. Then $\displaystyle \varphi^{-1}\circ b_t$ is the local flow of $\displaystyle (\varphi^{-1})_*X$ around p. Then

$\displaystyle L_{(\varphi^{-1})_*X}(\varphi^*\alpha)$=d/dt t=0 $\displaystyle ((\varphi^{-1}\circ b_t)^*(\varphi^*\alpha))_p$ = d/dt t=0 $\displaystyle ((b_t)^*\alpha)_p=(L_X\alpha)_p$

what seems to be wrong. What is wrong?

Thanks in advance!

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

the flow that induces $\displaystyle (\varphi^{-1})_* X$ is $\displaystyle \varphi^{-1} \circ b_t \circ \varphi$.

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

Thanks xxp9, I looked up the definition of $\displaystyle (\varphi^{-1})_* X$ and get the right solution now.