Petty projection inequality on the sphere and on the hyperbolic space

Using gnomonic projection and Poincaré model, we first define the spherical projection body and hyperbolic projection body in spherical space ${\mathbb{S}}^n$ and hyperbolic space ${\mathbb{H}}^n$, then define the spherical Steiner symmetrization and hyperbolic Steiner symmetrization, finally prove the spherical projection inequality and hyperbolic projection inequality.