If I had some arbitrary convex 3D object that that lives in H2xE hyperbolic space and project the two hyperbolic dimensions into a Beltrami–Klein cylinder space, would the projection also be convex?
For context in why I am asking: I want to simulate physical collisions in H2xE. My thought is to project the hyperbolic dimensions to Beltrami–Klein (which as I understand retains linearity), then perform the collision tests on the resulting shapes as if they were fully Euclidean. But this could only feasibly work if I had a guarantee that convex objects remain convex when projected.
No. The curve
$$x(t)=\tanh(at)\cos(\theta)$$ $$y(t)=\tanh(at)\sin(\theta)$$ $$z(t)=bt$$
represents a geodesic in $\mathbb H^2\times\mathbb E^1$, but not a straight line in $\mathbb E^3$ (unless $ab=0$).