Geometric decomposition of geodesics and null-phase curves using Majorana star representation

Geodesics are the shortest curves between any two points on a given surface. in state space of quantum systems play an important role theory geometric phases, as these also along which acquired phase is zero. Null (NPCs) generalization geodesics, defined zero even though they need not be points. Here we present decomposition geodesics and NPCs higher-dimensional space, allows understanding intrinsic symmetries curves. We use Majorana star representation to decompose geodesic $n$-dimensional Hilbert $n-1$ Bloch sphere show that all circular segments with specific properties determined by inner product end states connected geodesic. propose method construct infinitely many arbitrary for $(n>2)$-dimensional using our decomposition.