Are you familiar with how a vector field may be regarded as an infinitesimal diffeomorphism of the base space?
A delightful topic!

Yes, I'm familiar with the concept. In differential geometry, a vector field on a manifold $M$ can be regarded as an infinitesimal diffeomorphism of $M$. This is a fundamental idea that reveals a deep connection between vector fields and diffeomorphisms.

To elaborate, let $X$ be a vector field on $M$. We can associate with $X$ a one-parameter group of diffeomorphisms, $\phi_t: M \to M$, which is a family of diffeomorphisms depending on a parameter $t \in \mathbb{R}$. The idea is that $\phi_t$ represents the "flow" of the vector field $X$ over a time interval $t$.

The infinitesimal diffeomorphism corresponding to $X$ is the derivative of $\phi_t$ at $t=0$, denoted by $\left.\frac{d\phi_t}{dt}\right|_{t=0}$. This derivative is a linear map from the tangent space $T_pM$ at a point $p \in M$ to itself, which can be identified with the vector $X_p \in T_pM$.

In other words, the vector field $X$ can be thought of as an infinitesimal generator of a one-parameter group of diffeomorphisms, and the diffeomorphisms in this group are obtained by integrating the flow of $X$
  Yes, a vector field on a manifold can be regarded as an infinitesimal diffeomorphism of the base space. In fact, this is a fundamental way of understanding the geometry of a vector field on a manifold.

To explain this, let's first define what we mean by an infinitesimal diffeomorphism. Given a manifold $M$ and a point $p \in M$, an infinitesimal diffeomorphism of $M$ at $p$ is a map $f: T_pM \to T_pM$ such that $f$ is a linear transformation on the tangent space $T_pM$ at $p$, and such that $f$ satisfies the chain rule of differentiation with respect to the manifold's coordinate chart. In other words, if we have a smooth function $g: U \to \mathbb{R}$, where $U$ is an open set in $M$, and if we have a tangent vector $v \in T_pM$ to $M$ at $p$, then $f(v)$ is the tangent vector to the function $g \circ \phi$ at $p$, where $\phi: U \to U$ is a smooth chart on $M$ centered at $p$.

Now, let's consider a vector field $V$ on a manifold $M$. We can regard $V$ as an infinit