Bi-orientable maps (also called pseudo-orientable maps) were introduced by Wilson in the 1970s to describe non-orientable maps with the property that opposite orientations can consistently be assigned to adjacent vertices. In contrast to orientability, which is both a combinatorial and topological property, bi-orientability is only a combinatorial property. In this paper we classify the bi-orientable maps whose local-orientation-preserving automorphism groups act regularly on arcs, called here bi-rotary maps, of negative prime Euler characteristic. Unlike other classification results for highly symmetric maps on such surfaces, we do not use the Gorenstein–Walter result on the structure of groups with dihedral Sylow 2-subgroups.

In this paper we construct an infinite family of hypertopes of rank four having the complete graph $K_4$ as diagram. Their group of rotational symmetries is isomorphic to $PSL(2,q)$. It turns out some elements of this family are regular hypertopes and some are chiral. Moreover, we show that the chiral ones have both improper and proper correlations simultaneously.

We give here a geometric proof of the existence of certain local coordinates on a pseudo‐Riemannian manifold admitting a closed conformal vector field.