Motivation of Uhlenbeck’s trick without local computation(Suitable connection defined in space time manifold M $\times \mathbb{R}$)

Question

I am a beginner to learn Ricci flow and my main study reference is Simon Brendle’s “Ricci Flow and the Sphere Theorem”. In section 2.3 of Brendle, he introduce the following: let D be the Levi-Civita connection of $TM$ where $M$ is a Riemannian manifold and we define a new connection on $TM \times \mathbb{R}$ via \begin{equation} D_{\frac{\partial}{\partial t}}X = \frac{\partial}{\partial t}X- \sum_{k=1}^n \text{Ric}(X,e_k)e_k \quad(*) \end{equation} which is known as Uhlenbeck’s trick. Here $\{e_k\}_{k=1}^n$ is orthonormal frames of $TM$.

I just want to ask the motivation of (*). There are already similar questions: Linear connection other than Levi-Civita connection and Reference of Uhlenbeck’s trick, but neither of them can solve my question. Since Brendle prefers to do computation in tensor and avoid working in local coordinate, I am glad to hear some motivations of $(*)$ without using local computation.

Followed the advice of the second hyperlink above, I refer to Andrews and Hopper’s book The Ricci Flow in Riemannian Geometry. They discussed in lengthy that how to choose a suitable connection on space time manifold $M \times \mathbb{R}$ but I just couldn’t relate to $(*)$ directly.

The following are my attempts: the new connection should satisfy compatibility with Riemannian metric, thus

$$ \frac{\partial}{\partial t}g(X,e_l)= g(D_{\frac{\partial}{\partial t}}X, e_l)+0 $$ since $e_l$ is time-independent vector field. On the other hand, if we compute directly $$ \frac{\partial}{\partial t} g(X,e_l) = -2 \text{Ric}(X,e_l)+ g(\frac{\partial}{\partial t} X, e_l) $$ Here we use $\frac{\partial}{\partial t} g(X,Y) = -2 \text{Ric}(X,Y)$. Does this implies $$ \begin{equation} D_{\frac{\partial}{\partial t}}X = \frac{\partial}{\partial t}X- 2\sum_{k=1}^n \text{Ric}(X,e_k)e_k \quad(*) \end{equation} $$

IF the above attempts is true, why there is a factor 2 discrepancy with $(*)$? I hope to hear your insights, and any comments are welcome! Thank you in advance.

Answer 1

This is the introduction lines of Peter Topping Lectures on Ricci flow (sec 9.4):

We have arrived at a relatively simple evolution equation for $E$, and hence for the curvature. We wish to apply a maximum principle to find curvature conditions which are preserved under the Ricci flow. One remaining obstacle is that it will be important to see $E$ not just as a section of $T^*M\otimes TM$, but as a section of the sub-bundle of such sections which are symmetric with respect to $g(t)$. (That is, $g(E(X), Y ) = g(E(Y ),X)$ for all vector fields $X$ and $Y$.) Unfortunately, this sub-bundle inherits the $t$-dependence of $g(t)$, which would cause problems. we use the so-called ‘Uhlenbeck trick’ to hide this $t$-dependence.