My undergraduate thesis is related to the Ricci flow, and I have a number of basic questions.
Let $M$ be a smooth manifold.
At the start of Chapter 2.3 of Peter Topping's Lectures on the Ricci flow, he supposes that $g$ is a smooth family of Riemannian metrics on $M$ defined on an open interval.
Question 1. Does Peter mean that $g$ is a map of the form $g:I \times M \rightarrow T^{(0,2)}TM$, where $I\subseteq \mathbb R$ is an open interval? Does smoothness here mean that $g$ is smooth as a map between the smooth manifolds $I \times M$ and $T^{(0,2)}TM$?
I'm trying to write the definition of the Ricci flow in my own words. This is what I have: Let $I$ be an open interval, and let $g:I \times M \rightarrow T^{(0,2)}TM$ be a smooth (in the sense of Question 1) family of metrics. For each $p \in M$, we have a smooth curve $g(\cdot,p):I \rightarrow T^{(0,2)}T_p M$ given by $t \mapsto g(t,p)$. For each $t \in I$, define $(\partial_t g)(t,p) := g(\cdot,p)'(t)$. Since $T^{(0,2)}T_p M$ is a vector space, we can view $g(\cdot,p)'(t)$ as an element of $T^{(0,2)}T_p M$. We say that $g$ is a solution to the Ricci flow if
$$(\partial_t g)(t,p) = -2 \text{Ric}(g(t,p)) \qquad \forall (t,p) \in I \times M.$$
Question 2. Is my definition correct? Does the solution of the Ricci flow need to be smooth in the sense of Question 1?
Theorem A.15 of The Ricci Flow: Techniques and Applications Part I states the following:
Theorem 1. If $(M,g_0)$ is a closed Riemannian manifold, then there exists a unique solution $g(t)$ to the Ricci flow defined on some positive time interval $[0, \varepsilon)$ such that $g (0) = g_0$.
Question 3. In the theorem above, what does it mean for the solution $g:[0,\varepsilon) \times M \rightarrow T^{(0,2)}TM$ to be smooth? Does it mean that $g$ is smooth as a map between smooth manifolds, where $[0,\varepsilon) \times M$ is considered as a manifold with boundary? Or does it mean that $g$ is continuous, and smooth on $(0,\varepsilon) \times M$? Or does it mean something else?
On the other hand, the wikipedia page on the Ricci flow states that Hamilton proved the following theorem:
Theorem 2. If $(M,g_0)$ is a closed Riemannian manifold, then there exists a positive number $T$ and a Ricci flow $g_t$ parameterised by $t \in (0,T)$ such that $g_t$ converges to $g_0$ in the $C^\infty$ topology as $t \rightarrow 0$.
Question 4. What is the relationship between Theorem 1 and Theorem 2? Does one follow from the other?
I tried looking at the original paper by Hamilton, but I wasn't able to understand it.
Best Answer
Questions 1, 2: Yes, your explanations are correct. Since this description is rather abstract, I prefer to write it all out in local coordinates. Then the definitions are much more straightforward, because they involve just real-valued functions on an open subset of $\mathbb{R}^n$.
Q3: It is unfortunate that the initial definitions above assume an open interval. They all also hold for any type of nonempty connected interval. You don't need to define a manifold with boundary. If you write it all in local coordinates, then the Ricci flow looks like a map $$F: [a,b) \times O \rightarrow \mathbb{R}^N,$$ where $O \subset\mathbb{R}^n$ is open. Given what you've said, it's unclear to me what the assumptions are on $g_t$ in Theorem 1. It appears to be your second guess.
Q4: Yes, Theorem 1 is a consequence of Theorem 2. But it looks like Topping proves Theorem 1 first. Theorem 2 is not a direct consequence of Theorem 1. However, it is straightforward to apply the proof of Theorem 1 to not just the Ricci flow itself but also to the nonlinear heat equation satisfied by all of the covariant derivatives (with respect to the metric $g_0$) of $g_t$.