  With the growth of geometric science, including the methods of exploring the
world of information by means of modern geometry, there has always been a
mysterious and fascinating ambiguous link between geometric, topological and
dynamical characteristics with quantum entanglement. Since geometry studies the
interrelations between elements such as distance and curvature, it provides the
information sciences with powerful structures that yield practically useful and
understandable descriptions of integrable quantum systems. We explore here
these structures in a physical system of $N$ interaction spin-$1/2$ under
all-range Ising model. By performing the system dynamics, we determine the
Fubini-Study metric defining the relevant quantum state space. Applying
Gaussian curvature within the scope of the Gauss-Bonnet theorem, we proved that
the dynamics happens on a closed two-dimensional manifold having both a
dumbbell-shape structure and a spherical topology. The geometric and
topological phases appearing during the system evolution processes are
sufficiently discussed. Subsequently, we resolve the quantum brachistochrone
problem by achieving the time-optimal evolution. By restricting the whole
system to a two spin-$1/2$ system, we investigate the relevant entanglement
from two viewpoints; The first is of geometric nature and explores how the
entanglement level affects derived geometric structures such as the
Fubini-Study metric, the Gaussian curvature, and the geometric phase. The
second is of dynamic nature and addresses the entanglement effect on the
evolution speed and the related Fubini-Study distance. Further, depending on
the degree of entanglement, we resolve the quantum brachistochrone problem.
