A reduction correlation between the \((3+1)\) -dimensional variable-coefficient Gross–Pitaevskii equation with the partially nonlocal nonlinearity under a harmonic potential and a \((2+1)\) -dimensional constant-coefficient one is firstly erected. With the aid of solutions via the Hirota method for the \((2+1)\) -dimensional constant-coefficient equation, the \((3+1)\) -dimensional soliton analytical solutions with the Hermite–Gaussian envelope including vortex, diploe soliton and saddle-shaped soliton are firstly unfolded. Expanded and compressed evolutions of these \((3+1)\) -dimensional soliton structures are presented in the periodic amplification and exponential diffraction decreasing systems. In the \(x-z\) plane, the eye-shaped structure appears in all soliton structures, and its number is related to the Hermite parameter.