Vanishing curvature on a normal neighborhood is a local isometry to Minkowski space

(Jacobi fields are linear in the parallel transport.) Fix $v\in W$ and let $P_t:T_{p}M\to T_{{\gamma }_v(t)}M$ be the parallel transport along ${\gamma }_v$ from $0$ to $t$ for $t\in [0,1]$; by [L3], $P_t$ is a linear isometry and $t\mapsto P_{t}X$ is the unique parallel field along ${\gamma }_v$ with value $X$ at $t=0$. Since ${\gamma }_v([0,1])\subseteq {\exp }_p(W)=U$ by star-shapedness, $R({\gamma }_v^{\mathrm{\prime }}(t),\cdot ){\gamma }_v^{\mathrm{\prime }}(t)=0$ for every $t$, so $D_t^{2}J=0$ is the Jacobi equation of [L3]. The field $J(t):=t{P}_{t}X$ satisfies $J(0)=0$, $D_{t}J=P_{t}X$ (because $D_t(P_{t}X)=0$ and $D_t(t{P}_{t}X)=P_{t}X$), hence $D_t^{2}J=0$ and $D_{t}J(0)=X$. By uniqueness of Jacobi fields with prescribed initial data, $J_X(t)=t{P}_{t}X$.

(Converse direction.) Suppose a chart $(V,y)$ has constant components $g_{ab}={\eta }_{ab}$. Then all ${\mathrm{\partial }}_c{g}_{ab}$ vanish, and [L4] gives ${\mathrm{\Gamma }}^a{}_{bc}=0$ and $R^a{}_{bcd}=0$ on $V$; since the curvature is a tensor and its components vanish in one chart, $R\equiv 0$ on $V$. This proves (e).

(The differential of the exponential map is an isometry.) Fix $v\in W$ and $X,Y\in T_{p}M$. By [L2] and step 1.1, $d({\exp }_p{)}_v(X)=J_X(1)=P_{1}X$ and likewise $d({\exp }_p{)}_v(Y)=P_{1}Y$. The map $d({\exp }_p{)}_v$ is linear because it is a differential, and by the isometry property of $P_1$ in [L3], $g_{{\exp }_p(v)}(d({\exp }_p{)}_{v}X,d({\exp }_p{)}_{v}Y)=g_{{\exp }_p(v)}(P_{1}X,P_{1}Y)=g_p(X,Y)\mathrm{.}$ Hence $d({\exp }_p{)}_v$ is a linear isometry. This proves (a).

(The exponential map is a local isometry.) By [L1] the map ${\exp }_p{\mathrm{\mid }}_W:W\to U$ is a diffeomorphism onto $U$, and by step 2.1 its differential at every $v\in W$ is a linear isometry; by [L5] this means $({\exp }_p{)}^{\ast }g=g_p$ on $W$, that is, ${\exp }_p$ is an isometry of $(W,g_p)$ onto $(U,g)$. This proves (b).

(Constant components in the exponential chart.) Let $u=\psi \circ {\exp }_p^{-1}$ be the exponential chart and ${\mathrm{\partial }}_a^u$ the corresponding coordinate fields; by [L5] applied to the diffeomorphism ${\exp }_p$ and the definition $e_a={\psi }^{-1}$-image of the standard basis, $g_{ab}(q)=({\exp }_p^{\ast }g{)}_{{\exp }_p^{-1}(q)}(e_a,e_b)=g_p(e_a,e_b)={\eta }_{ab}(q\in U)\mathrm{.}$ Thus all components of $g$ in the chart $u$ equal the constants ${\eta }_{ab}$, proving (c) and the displayed formula of the statement; in particular the chart $u$ is an isometry of $U$ onto the open set $u(U)\subseteq {\mathbb{R}}^4$ carrying the Minkowski form.

(Open flat sets and conclusion.) Let $p\in O$ with $R\equiv 0$ on the open set $O$. Since ${\exp }_p$ is continuous with ${\exp }_p(0)=p$ and $O$ is open, there is $\delta >0$ with ${\exp }_p(B_{\delta }(0)\cap {\mathcal{E}}_p)\subseteq O$; the set $W_0:=B_{\delta }(0)\cap {\mathcal{E}}_p$ is open and star-shaped about $0$ (an intersection of star-shaped sets), and after shrinking $\delta$ so that ${\exp }_p$ is a diffeomorphism on $W_0$ the hypotheses of the theorem hold with $W_0$ in place of $W$, so conclusions (a)-(c) hold for $W_0$; this proves (d) for the point $p$, and applying the same construction at each $q\in O$ gives each point of $O$ a neighborhood isometric to an open subset of Minkowski space. Together with step 1.2 this proves (d) and (e), and steps 2.1, 3.1 and 4.1 prove (a), (b) and (c) under the stated hypothesis.