Let $M$ be a manifold and $U\subseteq \mathbb R$ an open interval containing $0$. Consider a smooth map $\Phi:U \times M\rightarrow M$ such that
- $\Phi(t,-):M\rightarrow M$ is a diffeomorphism, for every $t\in U$
- $\Phi(0,-)$ is the identity morphism.
Then, for every $x\in M$, we have a path $\Phi(-,x):(U,0)\rightarrow (M,x)$ representing a tangent vector $v_x=\Phi(-,x)_*(\frac{d}{dt})\in T_xM$ and so we get a vector field $v:M\rightarrow TM$. This vector field has itself a maximal flow $\Psi:V\rightarrow M$ (defined on some open $V\subseteq \mathbb R\times M$ containing $\{0\}\times M$) and so I define the following $$\mathcal D=\{t\in\mathbb R|(t,x)\in V\forall x\in M\}$$ so $\mathcal D$ is the maximal domain on which every maximal integral curve is defined. My question is
In the context of this question, is $\mathcal D$ an open neighborhood of $0$?
For those interested, here is where this question comes from: I'm reading about the internal tangent space to a diffeological space. I'm particularly interested in the tangent space at the identity of the diffeological space $\text{Diff}(M)$. The construction of the internal tangent space (ITS, for short) is here and here. In general, the ITS at some point $x$ of the diffeological space $X$ is linearly generated by pairs $(p,u)$, with $p: (W,0)\rightarrow(X,x)$ a plot for $X$ (with $W$ open neighborhood of $0\in\mathbb R$) and $u\in T_0W$. For every smooth $f:(W',0)\rightarrow (W,0)$ we identify $(pf,u)$ with $(p,f_*(u))$.
In here, the author proves that if $G$ is a diffeological group, then every tangent vector $v\in T_gG$ ($g$ any point $\in G$) can be represented by a single pair $(p,u)$ like above. In particular, this applies to $\text{Diff}(M)$, so every $v\in T_\text{id}\text{Diff}(M)$ is represented by some plot $p:U\rightarrow \text{Diff}(M)$ sending $0$ to $\text{id}$ and a tangent vector $u\in T_0U$. Now, a plot $p$ is like above a smooth family $U\times M\rightarrow M$ of diffeomorphisms, in particular given the pair $(p,u)$ we can associate a vector field as $$x\mapsto p(-,x)_*(u)\in T_xM$$so that we have a map $$\gamma:T_\text{id}\text{Diff}(M)\rightarrow \mathfrak X(M)$$ If $M$ is compact, then $\gamma$ is an isomorphism (proved in the articles above). In general, I want to understand the image of $\gamma$ for a non-necessarily compact manifold. For now, my guess is that the image of $\gamma$ are those vector fields for which $\mathcal D$ (defined as above) is an open neighborhood of $0$.