  In this paper, we address an extension of the theory of self-concordant
functions for a manifold. We formulate the self-concordance of a geodesically
convex function by a condition of the covariant derivative of its Hessian, and
verify that many of the analogous properties, such as the quadratic convergence
of Newton's method and the polynomial iteration complexity of the
path-following method, are naturally extended. However it is not known whether
a useful class of self-concordant functions/barriers really exists for
non-Euclidean manifolds. To this question, we provide a preliminary result that
the squared distance function in the hyperbolic space of curvature $- \kappa$
is $\sqrt{\kappa}/2$-self-concordant and the associated logarithmic barrier of
a ball of radius $R$ is an $O(\kappa R^2)$-self-concordant barrier. We also
give an application to the minimum enclosing ball in a hyperbolic space.
