Exercise 4, § 8

Let $G$ be a finite-dimensional real Lie group and $A$ a subgroup of $G$. We say that an element $x$ of $L(G)$ is $A$-accessible if, for every neighbourhood $U$ of $e$, there exists a continuous mapping $\alpha$ of $(0, 1)$ into $A$ such that $\alpha(0) = e$ and $\alpha(t) \in \exp(tx) \cdot U$ for $0 \leq t \leq 1$. Let $\mathfrak{h}$ be the set of $A$-accessible elements of $L(G)$.

(a) Show that $\mathfrak{h}$ is a Lie subalgebra of $L(G)$. (Use Exercise 19 of § 6.)

(b) Let $H$ be an integral subgroup of $G$ such that $L(H) = \mathfrak{h}$. Show that $H \subset A$. (Let $I = [-1, 1]$ and let $\mathbf{R}^r$ be given the Euclidean norm. Let $(x_1, \ldots, x_r)$ be a basis of $\mathfrak{h}$. Construct continuous mappings $\alpha_1, \ldots, \alpha_r$ of $I$ into $A$ and continuous mappings $f_1, \ldots, f_r$ from $I^r$ to $\mathbf{R}$, such that, for $t = (t_1, \ldots, t_r) \in I^r$,

Then apply the following theorem: let $f$ be a continuous mapping of $I^r$ into $\mathbf{R}^r$ such that $\| f(x) - x \| \leq \frac{1}{2}$ for all $x \in I^r$; then $f(I^r)$ contains a neighbourhood of 0 in $\mathbf{R}^r$.†)

† This follows from the Brouwer fixed point theorem, which can be consulted, for example, in N. Dunford and J. T. Schwartz, Linear operators, part I (Interscience publishers, 1958), pp. 467–470.

(c) Deduce that $\mathfrak{h}$ is the subalgebra tangent to $A$ at $e$. (d) Show that if $A$ is arcwise connected, then $A = H$.†