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With the growth of geometric science, including the methods of exploring theworld of information by means of modern geometry, there has always been amysterious and fascinating ambiguous link between geometric, topological anddynamical characteristics with quantum entanglement. Since geometry studies theinterrelations between elements such as distance and curvature, it provides theinformation sciences with powerful structures that yield practically useful andunderstandable descriptions of integrable quantum systems. We explore herethese structures in a physical system of $N$ interaction spin-$1/2$ underall-range Ising model. By performing the system dynamics, we determine theFubini-Study metric defining the relevant quantum state space. ApplyingGaussian curvature within the scope of the Gauss-Bonnet theorem, we proved thatthe dynamics happens on a closed two-dimensional manifold having both adumbbell-shape structure and a spherical topology. The geometric andtopological phases appearing during the system evolution processes aresufficiently discussed. Subsequently, we resolve the quantum brachistochroneproblem by achieving the time-optimal evolution. By restricting the wholesystem to a two spin-$1/2$ system, we investigate the relevant entanglementfrom two viewpoints; The first is of geometric nature and explores how theentanglement level affects derived geometric structures such as theFubini-Study metric, the Gaussian curvature, and the geometric phase. Thesecond is of dynamic nature and addresses the entanglement effect on theevolution speed and the related Fubini-Study distance. Further, depending onthe degree of entanglement, we resolve the quantum brachistochrone problem.

Complementarity between quantum entanglement, geometrical and dynamical appearances in $N$ spin-$1/2$ system under all-range Ising model