  Topology, a mathematical concept in geometry, has become an ideal theoretical
tool for describing topological states and phase transitions. Many topological
concepts have found their physical entities in real or reciprocal spaces
identified by topological/geometrical invariants, which are usually defined on
orientable surfaces such as torus and sphere. It is natural to quest whether it
is possible to find the physical realization of more intriguing non-orientable
surfaces. Herein, we show that the set of spin-induced ferroelectric
polarizations in cubic perovskite oxides AMn3Cr4O12 (A = La and Tb) resides on
the topological Roman surface, a non-orientable two-dimensional manifold formed
by sewing a Mobius strip edge to that of a disc. The induced polarization may
travel in a loop along the non-orientable Mobius strip or orientable disc
depending on how the spin evolves as controlled by external magnetic field.
Experimentally, the periodicity of polarization can be the same or the twice of
the rotating magnetic field, being well consistent with the orientability of
disc and Mobius strip, respectively. This path dependent topological
magnetoelectric effect presents a way to detect the global geometry of the
surface and deepens our understanding of topology in both mathematics and
physics
