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

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 $$$$D_{\frac{\partial}{\partial t}}X = \frac{\partial}{\partial t}X- \sum_{k=1}^n \text{Ric}(X,e_k)e_k \quad(*)$$$$ which is known as Uhlenbeck’s trick. Here $$\{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 $$$$D_{\frac{\partial}{\partial t}}X = \frac{\partial}{\partial t}X- 2\sum_{k=1}^n \text{Ric}(X,e_k)e_k \quad(*)$$$$

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.

• The computations doesn’t seem to consider that $e_l$ is an orthonormal frame that changes with time, when the base metric $g(t)$ is evolving as well. In the book where $e_l$ is replaced with $Y$ the equations are indeed balanced. Commented Dec 28, 2021 at 2:07

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.