Extremal charged black holes with a twisted extra dimension

We construct odd-dimensional extremal charged black hole solutions with a twisted ${\mathrm{S}}^1$ as an extra dimension on generalized Euclidean Taub-NUT spaces. There exists a null hypersurface where an expansion for an outgoing null geodesic congruence vanishes, then these spacetimes look like black holes. We show that the metrics admit ${\mathrm{C}}^0$ extension across the horizon, but some components of Riemann curvature diverge there if the dimension is higher than five. The singularity is relatively mild so that an observer along a free-fall geodesic can traverse the horizon. We also show solutions with a positive cosmological constant.