The geodesic initial-value problem and the exponential map of the metric-selected connection

(Chart geodesic system.) Fix a chart $(U,x)$; its domain is an open subset $x(U)\subseteq {\mathbb{R}}^4$, so $D:=x(U)\times {\mathbb{R}}^4$ is open in ${\mathbb{R}}^8$, and the field $F(x,w)=(w,-\mathrm{\Gamma }(x)[w,w])$ is smooth on $D$ because the Christoffel symbols are smooth. By [L1], a smooth curve in $U$ is a geodesic exactly when its coordinate lift solves $\dot{z}=F(z)$. Conversely, every solution $z=(y,w)$ with values in $D$ satisfies $y^{\mathrm{\prime }}=w$ and $y^{\mathrm{\prime }\mathrm{\prime }}=-\mathrm{\Gamma }(y)[y^{\mathrm{\prime }},y^{\mathrm{\prime }}]$, so $x^{-1}\circ y$ is the corresponding geodesic.

(Local flow and joint smoothness.) By [L2], near any $z_0\in D$ there is a local flow $\mathrm{\Phi }:(-h_0,h_0)\times Z_0\to D$ solving ${\mathrm{\partial }}_t\mathrm{\Phi }=F(\mathrm{\Phi })$ with $\mathrm{\Phi }(0,y)=y$. Apply [L3] to the jointly smooth parameter field $H(r,z,\lambda ):=\lambda F(z)$ on $\mathbb{R}\times D\times \mathbb{R}$, at $(r,z,\lambda )=(0,z_0,0)$. There are $a>0$, an open neighborhood $Z\subseteq Z_0$ of $z_0$, and $b>0$ such that its solutions $U(r,y,\lambda )$ exist on $[-a,a]$ for $(y,\lambda )\in Z\times (-b,b)$; at the fixed time $c:=a\mathrm{/}2>0$, the map $(y,\lambda )\mapsto U(c,y,\lambda )$ is smooth. Shrink $b$ so $cb<h_0$. For $0\le r\le c$ and $\mathrm{\mid }\lambda \mathrm{\mid }<b$, the curve $r\mapsto \mathrm{\Phi }(r\lambda ,y)$ is defined and satisfies $d\mathrm{\Phi }(r\lambda ,y)\mathrm{/}dr=\lambda F(\mathrm{\Phi }(r\lambda ,y))$ with initial value $y$. Uniqueness in [L3] gives $U(c,y,\lambda )=\mathrm{\Phi }(c\lambda ,y)$, including $\lambda =0$. Consequently $\mathrm{\Phi }(t,y)=U(c,y,t\mathrm{/}c)$ for $\mathrm{\mid }t\mathrm{\mid }<cb$ and $y\in Z$. The right side is a composition of smooth maps in the pair $(t,y)$, so, with $h:=cb$, the restriction $\mathrm{\Phi }:(-h,h)\times Z\to D$ is jointly smooth and remains the unique local solution through each $y\in Z$.

(Agreement of solutions.) Let ${\gamma }_1,{\gamma }_2$ be geodesics whose parameter intervals share a nondegenerate interval $J$ containing a parameter $t_0$ with ${\gamma }_1(t_0)={\gamma }_2(t_0)$ and ${\gamma }_1^{\mathrm{\prime }}(t_0)={\gamma }_2^{\mathrm{\prime }}(t_0)$. Consider the set of $t\in J$ at which both their positions and tangent vectors agree. It is closed because the two lifted curves $t\mapsto ({\gamma }_i(t),{\gamma }_i^{\mathrm{\prime }}(t))$ in the Hausdorff tangent bundle are continuous. It is relatively open because, in a chart about any agreement point, the position--velocity lifts solve the same first-order system and local uniqueness from [L3] makes them agree nearby, including a one-sided neighborhood when that point is an included endpoint. It is nonempty by the data at $t_0$, so connectedness of $J$ makes it all of $J$. Applying this to each compact subinterval of the common domain proves agreement wherever both are defined. In particular two solutions through $(p,v)$ with the same value and tangent at parameter 0 agree on their common domain.

(Maximal geodesic.) Fix $(p,v)\in TM$ and let $\mathcal{S}$ be the set of all geodesics $\gamma$ defined on an open interval containing $0$ with $\gamma (0)=p$ and ${\gamma }^{\mathrm{\prime }}(0)=v$; it is nonempty by [L2] applied in a chart around $p$ to the state $(x(p),v)$, and its parameter domains all contain $0$. Let $I_{p,v}$ be the union of those domains and define ${\gamma }_{p,v}(t):=\gamma (t)$ for any $\gamma \in \mathcal{S}$ whose domain contains $t$; by step 2.1 this is independent of the choice of $\gamma$. The union of intervals containing $0$ is an interval containing $0$, and $I_{p,v}$ is open: if $t_0\in I_{p,v}$, then $t_0$ lies in the domain of some $\gamma \in \mathcal{S}$, and [L2] applied at the state of $\gamma$ at $t_0$ extends a representative solution to a neighbourhood of $t_0$ which, by step 2.1, agrees with ${\gamma }_{p,v}$ on the overlap. Every geodesic in $\mathcal{S}$ is a restriction of this union. To include arbitrary interval domains, let $\eta :J\to M$ be any geodesic with the same initial data, allowing included endpoints as in Geodesic of an affine connection. At each included finite endpoint $e$ of $J$, smooth one-sided data give a position--velocity state $(\eta (e),{\eta }^{\mathrm{\prime }}(e))$. Step 1.2 gives a two-sided local solution through that state in a chart about $\eta (e)$; the one-sided coordinate lift of $\eta$ solves the same ODE, so uniqueness in [L3] makes them agree on their common one-sided interval. They therefore glue to a smooth geodesic across $e$. Opening the at most two included endpoints gives an extension $\widehat{\eta }$ on an open interval $\widehat{J}\supseteq J$ containing $0$, so $\widehat{\eta }\in \mathcal{S}$. Hence $J\subseteq I_{p,v}$ and $\eta ={\gamma }_{p,v}{\mathrm{\mid }}_J$. In particular no extension on any strictly larger interval exists, whether open, closed or half-open, and any other maximal geodesic with the data equals this one. This proves (a).

(Smoothness of one geodesic.) For $\gamma ={\gamma }_{p,v}$ and $t_0\in I_{p,v}$, choose a chart around $\gamma (t_0)$ and apply step 1.2 at the state $(\gamma (t_0),{\gamma }^{\mathrm{\prime }}(t_0))$; the coordinate lift of $\gamma$ agrees near $t_0$ with the smooth local flow of that state, so $t\mapsto {\gamma }_{p,v}(t)$ is smooth on $I_{p,v}$.

(Scaling.) Let $s\in \mathbb{R}$. For $s=0$, the constant curve at $p$ is a geodesic with initial data $(p,0)$ and is defined on all of $\mathbb{R}$ by [L4], so by step 3.1 it is ${\gamma }_{p,0}$, and $t\in I_{p,0}$ iff $0=0\cdot t\in I_{p,v}$ holds for all $t$, with equal values $p={\gamma }_{p,v}(0)$. For $s\mathrm{\ne }0$, put $J_s:=\{t:st\in I_{p,v}\}$. On $J_s$ the curve $t\mapsto {\gamma }_{p,v}(st)$ is a geodesic with initial data $(p,sv)$: if $v\mathrm{\ne }0$, its tangent is nowhere zero, since a geodesic whose tangent vanished once would, by uniqueness in step 2.1, be constant; then clause 4 of [L4] applies to the affine reparametrization $h(t)=st$. If $v=0$, the curve is constant and clause 4 of [L4] handles that case directly. The interval $J_s$ is maximal for these initial data: any geodesic extension on a strictly larger interval would, after the inverse affine change of parameter $t\mapsto t\mathrm{/}s$, extend ${\gamma }_{p,v}$ beyond $I_{p,v}$, contrary to step 3.1. Thus uniqueness of the maximal geodesic in step 3.1 gives $J_s=I_{p,sv}$ and ${\gamma }_{p,sv}(t)={\gamma }_{p,v}(st)$, proving (c).

(Openness and smooth dependence.) Let $t_0\in I_{p_0,v_0}$. If $t_0=0$, step 1.2 and the flow identity give smooth dependence near $(0,p_0,v_0)$. Otherwise let $K$ be the compact interval with endpoints $0,t_0$ and let $z:K\to TM$ be the position--velocity lift of ${\gamma }_{p_0,v_0}$. For every $s\in K$, step 1.2 supplies an open neighborhood $Z_s$ of $z(s)$ in a tangent-bundle chart and a local flow on $(-h_s,h_s)\times Z_s$ for some $h_s>0$. Compactness gives finitely many such neighborhoods $Z_1,\dots ,Z_m$ covering $z(K)$; their inverse images under $z$ form an open cover of $K$, so [L6] supplies a Lebesgue number $\delta >0$. Choose a partition of $K$ with mesh less than $\min (\delta ,h_1,\dots ,h_m)$, traversed from $0$ toward $t_0$. Each subinterval then lies in some $z^{-1}(Z_i)$ and has length less than $h_i$, so its solution map is the corresponding local flow map. Compose these finitely many smooth maps; at each stage continuity lets us shrink the neighborhood of $(p_0,v_0)$ so the nearby state at the next node remains in the next selected flow neighborhood. Thus the position--velocity state at the last node depends smoothly on $(p,v)$ nearby, and a final local flow around $z(t_0)$ gives smoothness jointly in $(t,p,v)$ near $(t_0,p_0,v_0)$. This neighborhood is contained in $\mathcal{G}$, so $\mathcal{G}$ is open in $\mathbb{R}\times TM$ and the evaluation is smooth, proving (b).

(The exponential domain.) By (c), $1\in I_{p,sv}$ iff $s\in I_{p,v}$ for every $s\in \mathbb{R}$. Hence ${\mathcal{E}}_p=\{v:1\in I_{p,v}\}$ is the preimage of the open set $\mathcal{G}$ under the continuous map $v\mapsto (1,p,v)$, so it is open in $T_{p}M$; and if $v\in {\mathcal{E}}_p$ and $0\le s\le 1$, then $s\in I_{p,v}$ because $I_{p,v}$ is an interval containing $0$ and $1$, so $1\in I_{p,sv}$ and $sv\in {\mathcal{E}}_p$: the domain is star-shaped about $0$. The map ${\exp }_p$ is the restriction of the smooth evaluation of (b) to $\{1\}\times \{p\}\times {\mathcal{E}}_p$, hence is smooth, and ${\exp }_p(0)={\gamma }_{p,0}(1)=p$.

(Differential at zero.) Choose a basis $e_0,\dots ,e_3$ of $T_{p}M$ with coordinate isomorphism ${\psi }_p:T_{p}M\to {\mathbb{R}}^4$ and identify $T_0(T_{p}M)$ with $T_{p}M$ by the chart ${\psi }_p$. For $w\in T_{p}M$, the curve $t\mapsto {\exp }_p(tw)$ has, by (c), ${\exp }_p(tw)={\gamma }_{p,tw}(1)={\gamma }_{p,w}(t)$, so its velocity at $0$ is ${\gamma }_{p,w}^{\mathrm{\prime }}(0)=w$; hence $d({\exp }_p{)}_0(w)=w$ for every $w$, that is $d({\exp }_p{)}_0={\mathrm{id}}_{T_{p}M}$.

(Local diffeomorphism.) Since $d({\exp }_p{)}_0$ is the identity, it is an isomorphism, so [L5] gives open neighborhoods $V\subseteq {\mathcal{E}}_p$ of $0$ and $U\subseteq M$ of $p$ such that ${\exp }_p{\mathrm{\mid }}_V:V\to U$ is a diffeomorphism. Choose a Euclidean norm on the finite-dimensional space $T_{p}M$ and let $W_p$ be the open ball of some radius contained in $V$; it is open, contains $0$, is star-shaped, and ${\exp }_p{\mathrm{\mid }}_{W_p}$ is a diffeomorphism onto its open image. This proves (d) and completes the proof.