Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points

(Setup and choice of $r$.) Fix the point-normal chart $(U,x)$ of [L1] and shrink it to satisfy [L2] with constants $r_0,\rho ,\mathrm{\Lambda },s_0$. Since $\mathrm{\Gamma }(p)=0$ by [L1] and the Christoffel symbols are continuous on the coordinate domain [L1], we may choose $r_1>0$ with $r_1\le r_0$ and $\mathrm{\parallel }\mathrm{\Gamma }{\mathrm{\parallel }}_{\overline{B_{r_1}}}{r}_1\le \frac{1}{100},r_1\le \frac{1}{4},x^{-1}(B_{r_1})\subseteq O\mathrm{.}$ Choose $r>0$ with $r\le \min \{\frac{r_1}{8{\mathrm{\Lambda }}^{\mathrm{\prime }\mathrm{\prime }}},\frac{s_0}{2},\frac{\rho }{32{\mathrm{\Lambda }}^{\mathrm{\prime }}},\frac{1}{100\max ({\mathrm{\Gamma }}_0{\mathrm{\Lambda }}^{\mathrm{\prime }},1)},\frac{1}{4}\},$ where ${\mathrm{\Lambda }}^{\mathrm{\prime }}:=\max (\mathrm{\Lambda },1)$, ${\mathrm{\Lambda }}^{\mathrm{\prime }\mathrm{\prime }}:=\max (\frac{8}{3}\mathrm{\Lambda },1)$ and ${\mathrm{\Gamma }}_0:=\mathrm{\parallel }\mathrm{\Gamma }{\mathrm{\parallel }}_{\overline{B_{r_1}}}$. Then $C=x^{-1}(B_r)\subseteq O\cap x^{-1}(B_{s_0})$.

(The two-point geodesic stays in the larger ball.) Let $q,q^{\mathrm{\prime }}\in \bar{C}$ and let ${\mathrm{\Gamma }}_0:=\mathrm{\parallel }\mathrm{\Gamma }{\mathrm{\parallel }}_{\overline{B_{r_1}}}$. Put $u:=v(q,q^{\mathrm{\prime }})\in T_{q}M$, so $\mathrm{\mid }u{\mathrm{\mid }}_x\le \mathrm{\Lambda }\mathrm{\mid }x(q)-x(q^{\mathrm{\prime }}){\mathrm{\mid }}_{\mathrm{e}}\le 2\mathrm{\Lambda }r\le \frac{\rho }{16}<\rho$ and $(q,u)\in V$; let $\gamma :[0,1]\to M$, $\gamma (t)={\exp }_q(tu)$, be the corresponding geodesic. Since $\mathrm{\mid }u{\mathrm{\mid }}_x\le 2\mathrm{\Lambda }r$ and $r\le \frac{1}{100\max ({\mathrm{\Gamma }}_0{\mathrm{\Lambda }}^{\mathrm{\prime }},1)}$, we have $4{\mathrm{\Gamma }}_0\mathrm{\mid }u{\mathrm{\mid }}_x\le 8{\mathrm{\Gamma }}_0\mathrm{\Lambda }r\le \frac{8}{100}<1$, so $\delta =1$ satisfies $4{\mathrm{\Gamma }}_0\mathrm{\mid }u{\mathrm{\mid }}_x\delta \le 1$; [L3] then gives, as long as the image stays in $x^{-1}(B_{r_1})$, $\mid \dot{x}(t){\mid }_{\mathrm{e}}\le \frac{4}{3}\mathrm{\mid }u{\mathrm{\mid }}_x\le \frac{8}{3}\mathrm{\Lambda }r,\mid x(\gamma (t))-x(q){\mid }_{\mathrm{e}}\le \frac{8}{3}\mathrm{\Lambda }r\mathrm{.}$ Since $\mathrm{\mid }x(q){\mathrm{\mid }}_{\mathrm{e}}\le r$ and $\frac{8}{3}\mathrm{\Lambda }r\le r_1\mathrm{/}8<r_1-r$ by the choice of $r$, an exit from $x^{-1}(B_{r_1})$ at the first parameter $T$ would require displacement at least $r_1-r$, while the speed estimate bounds it by $\frac{8}{3}\mathrm{\Lambda }r$. This is a contradiction; hence the whole image lies in $x^{-1}(B_{r_1})$.

(Interior containment for distinct endpoints.) By step 2.1 the image of $\gamma$ lies in $x^{-1}(B_{r_1})$. Its endpoints satisfy $\mathrm{\mid }x(q){\mathrm{\mid }}_{\mathrm{e}},\mathrm{\mid }x(q^{\mathrm{\prime }}){\mathrm{\mid }}_{\mathrm{e}}\le r$, and $\mathrm{\parallel }\mathrm{\Gamma }{\mathrm{\parallel }}_{\overline{B_{r_1}}}{r}_1\le \frac{1}{100}\le \frac{1}{2}$; [L4] applied with $r_2=r<r_1$ gives $\mathrm{\mid }x(\gamma (t)){\mathrm{\mid }}_{\mathrm{e}}\le r$ on $[0,1]$. If $q\mathrm{\ne }{q}^{\mathrm{\prime }}$ and $\mathrm{\mid }x(\gamma (t_0)){\mathrm{\mid }}_{\mathrm{e}}=r$ for some $t_0\in (0,1)$, the convex function $f(t)=\mathrm{\mid }x(\gamma (t)){\mathrm{\mid }}_{\mathrm{e}}^2$ attains its maximum $r^2$ at an interior point, hence is constant by [L4]. Then $f^{\mathrm{\prime }\mathrm{\prime }}\equiv 0$ while [L4] gives $f^{\mathrm{\prime }\mathrm{\prime }}\ge \mathrm{\mid }\dot{x}{\mathrm{\mid }}_{\mathrm{e}}^2$, so $\dot{x}\equiv 0$ and $\gamma$ is constant, contradicting $q\mathrm{\ne }{q}^{\mathrm{\prime }}$. Thus $\gamma ((0,1))\subset C$ for distinct endpoints. If $q=q^{\mathrm{\prime }}$, then $v(q,q)=0$ by [L2] and $\gamma$ is the constant geodesic.

(Uniqueness on the closure.) Let $\eta :[0,1]\to \bar{C}$ be any geodesic segment from $q$ to $q^{\mathrm{\prime }}$. No interior-containment assumption is needed for this uniqueness argument. Its image lies in $\overline{B_r}\subset x^{-1}(\overline{B_{r_1}})$. For every $r^{\mathrm{\prime }}\in (r,r_1)$, apply L3 with compact set $K=x^{-1}(\overline{B_{r_1}})$ and radius $r^{\mathrm{\prime }}$; its hypotheses hold because $r^{\mathrm{\prime }}<r_1\le 1\mathrm{/}4$ and ${\mathrm{\Gamma }}_0{r}^{\mathrm{\prime }}\le {\mathrm{\Gamma }}_0{r}_1\le 1\mathrm{/}100$. The image lies in $K\cap x^{-1}(B_{r^{\mathrm{\prime }}})$, so $\mathrm{\mid }{\dot{x}}_{\eta }(0){\mathrm{\mid }}_{\mathrm{e}}\le 4r^{\mathrm{\prime }}$. Letting $r^{\mathrm{\prime }}\downarrow r$ gives $\mathrm{\mid }{\dot{x}}_{\eta }(0){\mathrm{\mid }}_{\mathrm{e}}\le 4r\le \rho \mathrm{/}(8{\mathrm{\Lambda }}^{\mathrm{\prime }})<\rho$. Thus $(q,{\eta }^{\mathrm{\prime }}(0))\in V$ and $\mathrm{\Phi }(q,{\eta }^{\mathrm{\prime }}(0))=(q,q^{\mathrm{\prime }})=\mathrm{\Phi }(q,v(q,q^{\mathrm{\prime }}))$. Injectivity of $\mathrm{\Phi }{\mathrm{\mid }}_V$ gives ${\eta }^{\mathrm{\prime }}(0)=v(q,q^{\mathrm{\prime }})$, and uniqueness for the geodesic initial-value problem gives $\eta =\gamma$. This also excludes every nonconstant geodesic loop in $\bar{C}$ when $q=q^{\mathrm{\prime }}$, so the unique degenerate segment is constant. Together with step 3.1 this proves the closure-endpoint condition in the definition and the uniqueness assertion in (b).

(The domains $W_q$.) Fix $q\in C$ and let $W_q$ be as in (a). It is star-shaped by construction: if $u\in W_q$ and $\lambda \in [0,1]$, the segment $t\mapsto {\exp }_q(t\lambda u)$, $t\in [0,1]$, is the initial arc of $t\mapsto {\exp }_q(tu)$ by the scaling rule of The geodesic initial-value problem and the exponential map of the metric-selected connection clause (c), hence lies in $C$ and $\lambda u\in W_q$. It is open: if $u\in W_q$, the geodesic ${\gamma }_{q,u}$ is defined on $[0,1]$ with image in the open set $C$; choosing $\epsilon >0$ with ${\gamma }_{q,u}([0,1+\epsilon ])\subset C$ (possible by [L5] since ${\gamma }_{q,u}(1)\in C$ is an interior point and the geodesic extends) and using smooth dependence on initial data [L5], all $u^{\mathrm{\prime }}$ sufficiently close to $u$ have ${\gamma }_{q,u^{\mathrm{\prime }}}([0,1+\epsilon ])\subset C$, so $u^{\mathrm{\prime }}\in W_q$. And ${\exp }_q(W_q)=C$: the inclusion $\subseteq$ is by definition of $W_q$; for $q^{\mathrm{\prime }}\in C$ the velocity $v(q,q^{\mathrm{\prime }})$ lies in $W_q$ by step 3.1 and ${\exp }_q(v(q,q^{\mathrm{\prime }}))=q^{\mathrm{\prime }}$.

(${\exp }_q:W_q\to C$ is a diffeomorphism.) It is smooth because ${\exp }_q$ is smooth on its domain [L5]; it is surjective by step 4.2; it is injective: if ${\exp }_q(u)={\exp }_q(u^{\mathrm{\prime }})=q^{\mathrm{\prime }}$ with $u,u^{\mathrm{\prime }}\in W_q$, then the segments of $u$ and $u^{\mathrm{\prime }}$ lie in $C$, so [L3] gives $\mathrm{\mid }u{\mathrm{\mid }}_x,\mathrm{\mid }{u}^{\mathrm{\prime }}{\mathrm{\mid }}_x\le 4r<\rho$; hence $(q,u),(q,u^{\mathrm{\prime }})\in V$ and $\mathrm{\Phi }(q,u)=(q,q^{\mathrm{\prime }})=\mathrm{\Phi }(q,u^{\mathrm{\prime }})$ forces $u=u^{\mathrm{\prime }}$. Finally, for $u\in W_q$ we have $\mathrm{\mid }u{\mathrm{\mid }}_x\le 4r<\rho$ and $q\in x^{-1}(B_r)\subseteq x^{-1}(B_{s_0})$, so $(q,u)\in V$ and $d({\exp }_q{)}_u$ is invertible by [L5]. A bijective local diffeomorphism onto $C$ is a diffeomorphism by [L6]. This proves (a).

(Conclusion.) Steps 2.1, 3.1 and 4.1 establish (b); steps 4.2 and 5.1 establish (a); together with the definitions of Normal neighborhoods, convex normal sets, the position vector and the squared-displacement function this gives (c). This completes the proof.