I have stumbled upon the definition of halfspace in model geometry i.e $S^n$ with the standard intrinsic metric, $\mathbb{R}^n$ with the standard metric and the Hyperbolic space $\mathbb{H}^n$ with the canonical metric.
I assume by that (up to isometry) the following things were meant: The upper hemisphere in $\mathbb{S}^n$, the upper half plane in $\mathbb{R}^n$, the right quadrant in the upper half-plane model of $\mathbb{H}^n$.
I have not been able to find a definition for half space of a Riemannian manifold written down anywhere. Do you know of any reference?