Find $T_\mathrm{id}\left(\mathrm{Diff}(S^1)\right)$

Question

We established on last tutorial that $T_\mathrm{id}(\mathrm{Diff}(S^1))$ are vector fields on $S^1$. I'd be grateful for any explanation (formal or intuitive) standing behind this answer.

Answer 1

A one-parameter subgroup of ${\rm Diff}({\cal M})$ would amount to a smoothly varying deformation of the manifold, during which each point of $\cal M$ will trace out some path: to each tangent vector of $T_e{\rm Diff}({\cal M})$ we can associate the collection of tangent vectors of the paths each individual point traces out.

In response to your comments:

(1) "One-parameter subgroup" is a standard vocabulary term. When you speak of "depending on an angle" I suspect you're confusing ${\rm Diff}(S^1)$ with $S^1$. Forget the group structure on $S^1$.

(2) It should be obvious from context what I mean by deformation: think of the manifold as a sheet (of potentially higher dimension) that can be stretched or twisted (the fabric itself, that is - the space the material actually occupies remains unaltered). For instance, when we mentally picture rigid motions of the plane, we visualize them as animations, so in other words a family of affine maps that vary smoothly with time $t$. Hence, a one-parameter subgroup of diffeomorphisms. (This helpfully aids us in picturing the orientation-preserving morphisms as the connected component of a group of morphisms.)

The phrase "would amounts to" means exactly what it does in the English language as usual; equivalent phrases I could have used are "is effectively" or "can be thought of as."

(3) A tangent vector is a path through a given point (or more accurately, an equivalence class of paths, as different paths can represent the same tangent vector - see the Wikipedia article). Any tangent vector of the identity of ${\rm Diff}(S^1)$ can be represented by a one-parameter subgroup, which is in term a smoothly varying family of diffeomorphisms of $S^1$, an "animation" or "deformation" as I have already called it.

Given any path $\rho:I\to{\rm Diff}({\cal M}):t\mapsto \rho_{\large t}$ and point $p\in{\cal M}$, the deformation moves the point $p$ along its own path, but this time in the space $\cal M$ (remember to keep $\cal M$ and ${\rm Diff}({\cal M})$ separate). Namely, the path is given by $t\mapsto \rho_{\large t}(p)$. Remember paths represent tangent vectors, so this means $\rho$ induces a tangent vector at $p$. Thus, it induces a tangent vector at every point on the manifold, and in a smoothly varying way. That is, it induces a vector field. But if the path $\rho:I\to{\rm Diff}(M)$ starts at $e\in{\rm Diff}({\cal M})$, then it also represents a tangent vector, namely one of $T_e{\rm Diff}({\cal M})$. Thus we have established an assignment of vector fields to tangent vectors in $T_e{\rm Diff}({\cal M})$.

Conversely, given a vector field, one can construct its vector flow, which is a one-parameter subgroup of ${\rm Diff}({\cal M})$ and will represent the same tangent vector as whatever path we began with. That is to say, the correspondence between tangent vectors of the identity diffeomorphism and vector fields of the space is bijective. (Indeed it is an isomorphism of so-called Lie algebras.)

If you're comfortable with visualizing a single point moving along a trajectory and having an initial velocity vector, then it really shouldn't be a stretch to imagine that if every point moves along a trajectory (under some warping of the space, formally a smoothly varying family of diffomorphisms), then every point gets a tangent vector. Imagine atoms in the atmosphere as points, and the weather evolving over time as a smoothly varying family of self-maps of the atmosphere (a space of points) - here, a self-map at time $t$ tells us where atoms at time $t$ will be relative to their starting point. When you look at the weather channel and the map they show, they show tangent vectors (arrows) that tell you how all of the points are moving throughout space. So even utter nonmathematicians understand this idea intuitively!

Since a tangent vector tells you where an individual point is heading on an infinitessimal scale (intuitively speaking), a vector field tells us where all points are flowing under some smoothly varying family of diffeomorphisms. This is the justification of Qiaochu's excellent comment.