Assume that a Riemannian manifold $M$ is simply connected. In further, assume that there is an global orthnormal frame $e_i$. If $f_i$ is flow of $e_i$, and $M$ is homeomorphic to $\mathbb{R}^n$, then define $$ F(x)= f_n(x_n,\cdots f_2(x_2,f_1(x_1,o))\cdots ) $$ where $o$ is a fixed point. Then $F$ is a diffeomorphism from $\mathbb{R}^n$ to $M$. Hence an example of such manifold is a Lie group homeomorphic to $\mathbb{R}^n$, i.e., Heisenberg group $\mathbb{H}^3$.

Question : If Riemannian metric on Heisenberg group $M=\mathbb{H}^3$ has global orthonormal frame, then an exponential map ${\rm exp}$ is a diffeomorphism on $T_oM$, where $o$ is any fixed point.

Proof : We have a claim that between two points, there is unique geodesic, i.e. geodesic space.

Try 1 : At each point in $M$, sectional curvatures have both signs. If we have a point of negative sectional curvature, then it is helpful : There is no subset $A$ of measure $0$ in canonical sphere $S$ s.t. a connected set $S-A$ is a geodesic. But in some torus in $\mathbb{E}^3$ there is such property.

Try 2 : $M\rightarrow \mathbb{E}^2$ is a Riemannian submersion. So $\exp_p\ tv,\ \exp_p\ tw$ do not meet when $v,\ w$ are horizontal.

How can we finish the proof ?


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