Tangent space of a group of diffeomorphisms

Question

In a paper I was reading the following result was used: Let $\Gamma= Diff^{+}([0,1]^2)$ be the set of all boundary preserving diffeomorphisms on $[0,1]^2$, then the Tangent space $\mathcal{T}_{\gamma_{id}}(\Gamma)$ of $\Gamma$ at the identity $\gamma_{id}$ has the following form:$$\mathcal{T}_{\gamma_{id}}(\Gamma)=\{b:[0,1]^2\to [0,1]^2|b^{(1)}, b^{(2)}\in C^{\infty}([0,1]), \langle b,\gamma_{if}\rangle = 0\}.$$ I applied the defintions of a tangent space on this case, but I did not even get close to the given form. Can someone please show me how to proof this or give a literature reference where I can find something about this subject.

Answer 1

Here is the intuition, making this formal is hard because we are working with objects that in a sense are "infinite-dimensional manifolds". For example, even S. K. Donaldson in his article Moment maps and diffeomorphisms did not make an attempt to make all the objects formally well-defined.

Take a "smooth" path $$\phi:(-\epsilon, \epsilon)\longrightarrow\mathrm{Diff}^+([0, 1]^2)$$ with $\phi(0) = \mathrm{id}$. Then we have $$b(x, y) = \left.\frac{d}{dt}\right|_{t=0}\phi_t(x, y)\in T_{\phi_0(x, y)}[0, 1]^2 = T_{(x, y)}[0, 1]^2\cong\mathbb{R}^2,$$ giving you the fact that en element if the tangent space is a tangent field on $[0, 1]$.

The fact that we are working with boundary-preserving diffeomorphisms means that $\phi_t=\mathrm{id}$ on the boundary, so that we get the additional condition that $$b|_{\partial[0, 1]^2} = 0\ .$$

I am not convinced about the last condition in your definition. If it is correct, it should come from the imposition of the fact that $\phi_t$ must be invertible at all $t$, but I haven't been able to make that pop out for now.