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

Dirac equation for sphercially symmetric $AdS_5$ space-time and application to a boson star in EGB gravity