Since the rigidity theorem yields that the event horizon is a Killing horizon, the notion of surface gravity is well-defined. Then, by [*finite temperature*]{} we mean that the event horizon is of non-degenerate type, having non-vanishing surface gravity and a bifurcate surface [@RW92; @RW96]. For extremal (zero-temperature) black holes with vanishing surface gravity, the event horizon is of degenerated type and does not possess a bifurcate surface. Then, our boundary conditions to be imposed on target space fields at the event horizon would not appear to straightforwardly apply to such a case that the horizon has no bifurcate surface. It would be of great interest to consider the classification problem of such extremal (zero-temperature) black holes. In this respect, there have recently appeared some attempts to classify near-horizon geometries of extremal black objects, rather than extremal black objects themselves (see e.g., [@KL08a; @KL08b; @FKLR08; @KLR07; @AGJST06; @AY08] and references therein).

We also need to additionally impose the topology-condition (2), in order to explicitly specify boundary conditions on target space variables at the event horizon, in terms of certain coordinates, globally defined over the black hole exterior region. The topological censorship, together with our assumption of asymptotic flatness described above, immediately 