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We present a flexible scheme to realize exact flat Landau levels on curvedspherical geometry in a system of spinful cold atoms. This is achieved byFloquet engineering of a magnetic quadrupole field. We show that a syntheticmonopole field in real space can be created. We prove that the system can beexactly mapped to the electron-monopole system on sphere, thus realizingHaldane's spherical geometry for fractional quantum Hall physics. The schemeworks for either bosons or fermions. We investigate the ground state vortexpattern for an $s$-wave interacting atomic condensate by mapping this system tothe classical Thompson's problem. We further study the distortion and stabilityof the vortex pattern when dipolar interaction is present. Our scheme iscompatible with current experimental setup, and may serve as a promising routeof investigating quantum Hall physics and exotic spinor vortex matter on curvedspace.

Synthetic Landau levels and spinor vortex matter on Haldane spherical surface with magnetic monopole