If $\triangle$ is the diagonal of $X \times X$, show that its tangent space $T_{(x,x)}(\triangle)$ is the diagonal of $T_x(X) \times T_x(X).$ Is the following proof legit?
- $T_{(x,x)} \Delta \subseteq \Delta \subseteq T_x X \times T_x X$
All tangent vectors at the point $(x,x)$ in $\Delta$ are described by velocity vectors of curves passing through $(x,x)$. Suppose $c(t)$ is a curve through $(x,x)$ such that $c(0) = (x,x)$, then the velocity vector is given by $c'(0)$. Any $c(t)$ that lies in the diagonal looks like $c(t) = (\gamma(t), \gamma(t))$, where $\gamma(t)$ is a curve in $X$ through the point $x$ i.e., $\gamma(0) = x$ and $\gamma'(0) = v \in T_x X$. Then $c'(t) = (\gamma'(t), \gamma'(t))$ and $c'(0) = (v,v) \in \Delta \subseteq T_x X \times T_x X$.
- $T_x X \times T_x X \subseteq \Delta \subseteq T_{(x,x)} \Delta$
All tangent vectors at the point $x$ in $X$ are described by velocity vectors of curves passing through $x$. Suppose $r(t)$ is a curve through $x$ such that $r(0) = x$, then the velocity vector is given by $r'(0)$. Any element that lies in $T_x \times T_x$ looks like $(r^\prime(t), r^\prime(t))$, which is element in $T_{x,x}(\Delta)$.
Thanks.
This question intends to fill in the answer for the question GP 1.2.10(b) The tangent space $T_{(x,x)}(\triangle)$ is the diagonal of $T_x(X) \times T_x(X).$ And I do realize this is a very popular question, here's another identical problem again with partial solution: Show that the tangent space of the diagonal is the diagonal of the product of tangent space.