Formal vector fields vs. (standard) vector fields

Question

Given a smooth manifold $M$, one can consider the Lie algebra $\mathcal{X}(M)$ of vector fields equipped with the standard Lie bracket. This is a standard machinery of differential geometry. Gelfand and Fuchs defined the Lie algebra of formal vector fields at $0 \in \mathbb{R}^n$ as linear combinations $$\sum_{j=1}^np_j(x_1,...,x_n)e_j$$ where $e_j$ is a standard basis of $\mathbb{R}^n$ and $p_j$ are formal power series in variables $x_1,...,x_n$. This definition is at the given point $0$ in $\mathbb{R}^n$, so formal vector fields are not ,,globally'' defined.

Is it possible to define formal vector fields globally on a given manifold $M$?

Also I would like to know

What is the significance of formal vector fields? For example, do they naturally arise as a Lie algebra of some natural (infinite dimensional) group?

Answer 1

The Lie algebra of all formal vector fields of a real or complex manifold germ $(M,p)$ (where $p\in M$ is any point), does arise as the Lie algebra of the Lie group of all formal power series automorphisms of $(M,p)$, which is the same as the group of all invertible infinite jets of local diffeomorphisms of $(M,p)$, smooth or holomorphic respectively.

Also, formal vector fields of $(M,p)$ can be defined as derivations of the algebra of all formal power series at $p$.

Yet alternatively, the formal vector field algebra can be seen as the quotient of the algebra of all germs of smooth vector fields at $p$ modulo all germs of "flat" vector fields, i.e. ones vanishing of infinite order. For a complex manifold, that definition needs to be appropriately modified by allowing germs of smooth vector fields that are holomorphic of infinite order at $p$.

From that perspective, all these notions are fundamentally local. Of course, you can always consider subalgebras of formal vector fields extendible to global ones on $M$, which would be only interesting on a complex manifold. On a real manifold, of course, any formal vector field would arise that way.