Cartan hadamard

The exponential map is a local diffeomorphism and a local isometry for the pulled-back metric. [F1, F2, F5, given] If $v\mathrm{\ne }0$ and $\gamma (t)={\exp }_p(tv)$ had a conjugate pair $(0,1)$, then conjugacy and its multiplicity are unchanged under affine reparametrization (Conjugate points and multiplicity are invariant under affine reparametrization), so the unit-speed reparametrization of $\gamma$ would carry a nonzero Jacobi field vanishing at the start and at a later positive instant, contradicting [F1]; hence by [F2] the differential $d({\exp }_p{)}_v$ is invertible for every $v\mathrm{\ne }0$, and at $v=0$ the same holds by [F2]. Thus ${\exp }_p$ is a local diffeomorphism on all of $T_{p}M$, in particular an immersion, and $\tilde{g}:={\exp }_p^{\ast }g$ is a Riemannian metric on $T_{p}M$ by [F5]. By construction ${\exp }_p:(T_{p}M,\tilde{g})\to (M,g)$ satisfies $({\exp }_p{)}^{\ast }g=\tilde{g}$, so it is a local isometry, and the intertwining property of [F4] applies to it.

The straight rays through $0$ are $\tilde{g}$-geodesics defined for all time. [F3, F4, step 1.1] Fix $v\in T_{p}M$ and let $c(t):=tv$ be the straight ray in $T_{p}M$; the curve $c$ is defined for all $t\in \mathbb{R}$. Its image under ${\exp }_p$ is $({\exp }_p\circ c)(t)={\exp }_p(tv)={\gamma }_v(t)$, the $g$-geodesic of $M$ with ${\gamma }_v(0)=p$, ${\gamma }_v^{\mathrm{\prime }}(0)=v$, which by [F3] is defined for all real $t$ because $M$ is complete. Since $D_t^g({\exp }_p\circ c{)}^{\mathrm{\prime }}=0$ and by step 1.1 ${\exp }_p$ is a local isometry, [F4] gives $d({\exp }_p{)}_{tv}(D_t^{\tilde{g}}{c}^{\mathrm{\prime }})=D_t^g({\exp }_p\circ c{)}^{\mathrm{\prime }}=0$; the differential of the local diffeomorphism ${\exp }_p$ at $tv$ is invertible, so $D_t^{\tilde{g}}{c}^{\mathrm{\prime }}=0$ and $c$ is a $\tilde{g}$-geodesic. This holds for every $v\in T_{p}M$; in particular the straight ray begins at $0$ at time $0$, so the geodesic in $(T_{p}M,\tilde{g})$ with initial data $(0,u)$ is $t\mapsto tu$.

The pulled-back metric is complete. [F3, step 2.1] At the point $0\in T_{p}M$ the exponential map of the Riemannian manifold $(T_{p}M,\tilde{g})$ is defined on all of its tangent space: by step 2.1 the maximal $\tilde{g}$-geodesic with initial data $(0,u)$ is $t\mapsto tu$, defined for every $t\in \mathbb{R}$, so ${\exp }_0^{(\tilde{g})}(u)=u$ under the canonical identification $T_0(T_{p}M)\cong T_{p}M$. Thus ${\exp }_0^{(\tilde{g})}$ is globally defined (it is the identity of $T_{p}M$). The manifold $(T_{p}M,\tilde{g})$ is connected and boundaryless, so the equivalence of [F3] applied to it yields that $(T_{p}M,\tilde{g})$ is geodesically and metrically complete.

The exponential map is a covering map. [F6, step 1.1, step 3.1] The map ${\exp }_p:(T_{p}M,\tilde{g})\to (M,g)$ is a local isometry by step 1.1 between connected boundaryless Riemannian manifolds, and the source $(T_{p}M,\tilde{g})$ is nonempty and complete by step 3.1. Hence [F6] applies: ${\exp }_p$ is surjective and a smooth covering map.

Universal cover and the simply connected case. [F7, F8, step 4.1] If $\dim M=0$, then $T_{p}M=\{0\}$ and ${\exp }_p$ is the identity of a point, which is a diffeomorphism and trivially a universal cover. Otherwise $T_{p}M\cong {\mathbb{R}}^n$ is contractible by [F7], hence path connected with trivial fundamental group, hence simply connected by [F7]. By step 4.1 and the definition of a universal cover in [F6], ${\exp }_p$ is a universal cover of $M$ whenever $M$ is connected. If $M$ is simply connected as well, then $M$ is a locally path-connected simply connected space by [F8], so [F8] applies to the connected covering ${\exp }_p$: it is one-sheeted and isomorphic to the identity covering, in particular bijective. A bijective local diffeomorphism is a diffeomorphism, so ${\exp }_p$ is a diffeomorphism, and through it $M$ is diffeomorphic to the Euclidean space $T_{p}M$. Both assertions hold for every $p\in M$; the completeness of $(T_{p}M,\tilde{g})$ was proved, not assumed. The only selections are those made one at a time by the Hopf-Rinow and covering suppliers [F3] and [F6], so the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ of [A1] is consumed exactly through them.