In his book *"The geometry of geodesics"* H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms:

- The space is metric
- The space is finitely compact, i.e., a bounded infinite set has at least one accumulation point
- [
*metric convexity*] For every $x\neq z$ there exists a third point $y$ different from $x$ and $z$ such that $d(x,y)+d(y,z)=d(x,z)$ - [
*local prolongation*] To every point $p$ there corresponds $\rho_p>0$ such that for every two point $x,y\in S(p,\rho_p)$ there exists a point $z$ such that $d(x,y)+d(y,z)=d(x,z)$ - [
*uniqueness of prolongation*] If $d(x,y)+d(y,z_1)=d(x,z_1)$ and $d(x,y)+d(y,z_2)=d(x,z_2)$ and $d(y,z_1)=d(y,z_2)$ then $z_1=z_2$.

Busemann conjectured that every $G$-space is a topological manifold. My question is does every topological/smooth/Riemannian manifold is also a $G$-space?

As for connected complete Riemannian manifold, I figured out that **1** holds since by the metric. **3** holds since every two points can be joined by a minimal geodesic, and then we can pick $y$ to be a point on it. **4** holds since it is a manifold and locally it is homeomorphic to some Euclidean space. Unfortunately, even in this case, I couldn't figure out **5** and **2**.