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I call Poincaré $n$-half-space group the semidirect product of $\mathbb{R}^{n-1}$ and $\mathbb{R}^+$, where the action is by homotheties; equivalently as the group of translations and positive homotheties of $\mathbb{R}^{n-1}$. It is well-known that some left-invariant Riemannian metric on it is isometric to the $n$-dimensional hyperbolic space, i.e., has constant curvature $-1$.

What is an example of a Lie group $G$ not isomorphic to the Poincaré $n$-half-space group for any $n$, but satisfying the following:

 

For every left invariant metric $g$ on $G$, there exists a conformal rescaling $h$ of $g$ such that at each point $x\in G$ all $g$-geodesic rays initiating from $x$ have finite $h$-length, except exactly a single one.

The latter property is satisfied in the upper half plane, the conformal rescaling being the Euclidean metric of $\mathbb{R}^n$, and the exceptional direction being (at any point) the positive vertical one.

I call Poincaré $n$-half-space group the semidirect product of $\mathbb{R}^{n-1}$ and $\mathbb{R}^+$, where the action is by homotheties; equivalently as the group of translations and positive homotheties of $\mathbb{R}^{n-1}$. It is well-known that some left-invariant Riemannian metric on it is isometric to the $n$-dimensional hyperbolic space, i.e., has constant curvature $-1$.

What is an example of a Lie group $G$ not isomorphic to the Poincaré $n$-half-space group for any $n$, but satisfying the following:

For every left invariant metric $g$ on $G$, there exists a conformal rescaling $h$ of $g$ such that at each point $x\in G$ all $g$-geodesic rays initiating from $x$ have finite $h$-length, except exactly a single one.

The latter property is satisfied in the upper half plane, the conformal rescaling being the Euclidean metric of $\mathbb{R}^n$, and the exceptional direction being (at any point) the positive vertical one.

What is an example of a Lie group $G$ not ismorphic to the Poincare upper half space $H^n$ but satisfy the following:

For every left invariant metric $g$ we draw all geodesics. Then we CAN rescale $g$ conformally to a metric $h$ and we observe that at each point $x\in G$ all drown half ray $g$- geodesics initiating $x$ have finite $h$-length except a geodesic tangent to a unique direction

I call Poincaré $n$-half-space group the semidirect product of $\mathbb{R}^{n-1}$ and $\mathbb{R}^+$, where the action is by homotheties; equivalently as the group of translations and positive homotheties of $\mathbb{R}^{n-1}$. It is well-known that some left-invariant Riemannian metric on it is isometric to the $n$-dimensional hyperbolic space, i.e., has constant curvature $-1$.

What is an example of a Lie group $G$ not isomorphic to the Poincaré $n$-half-space group for any $n$, but satisfying the following:

For every left invariant metric $g$ on $G$, there exists a conformal rescaling $h$ of $g$ such that at each point $x\in G$ all $g$-geodesic rays initiating from $x$ have finite $h$-length, except exactly a single one.

The latter property is satisfied in the upper half plane, the conformal rescaling being the Euclidean metric of $\mathbb{R}^n$, and the exceptional direction being (at any point) the positive vertical one.

What is an example of a Lie group $G$ not ismorphic to the Poincare upper half space $H^n$ but satisfy the following:

For every left invariant metric $g$ we draw all geodesics. Then we rescale $g$ conformally to a metric $h$ and we observe that at each point $x\in G$ all drown half ray $g$- geodesics initiating $x$ have finite $h$-length except a geodesic tangent to a unique direction

What is an example of a Lie group $G$ not ismorphic to the Poincare upper half space $H^n$ but satisfy the following:

For every left invariant metric $g$ we draw all geodesics. Then we CAN rescale $g$ conformally to a metric $h$ and we observe that at each point $x\in G$ all drown half ray $g$- geodesics initiating $x$ have finite $h$-length except a geodesic tangent to a unique direction