  We discuss the Dirac equation in a curved 5-dimensional spherically symmetric
space-time. The angular part of the solutions is thoroughly studied, in a
formulation suited for extending to rotating space-times with equal angular
momenta. It has a symmetry $SU(2)\times U(1)$ and is implemented by the Wigner
functions. The radial part forms a Dirac-Schr\"odinger type equation, and
existence of the analytical solutions of the massless and the massive modes is
confirmed. The solutions are described by the Jacobi polynomials. Also, the
spinor of the both large and small components is obtained numerically. As a
direct application of our formulation, we evaluate the spectrum of the Dirac
fermion in Einstein-Gauss-Bonnet space-time and the space-time of a boson star.
