  The interplay between matter particles and gauge fields in physical spaces
with nontrivial geometries can lead to novel topological quantum matter.
However, detailed microscopic mechanisms are often obscure, and unconventional
spaces are generally challenging to construct in solids. Highly controllable
atomic systems can quantum simulate such physics, even those inaccessible in
other platforms. Here, we realize a Bose-Einstein condensate (BEC) on a
synthetic cylindrical surface subject to a net radial synthetic magnetic flux.
We observe a symmetry-protected topological band structure emerging on this
Hall cylinder but disappearing in the planar counterpart. BEC's transport
observed as Bloch oscillations in the band structure is analogous to traveling
on a M\"obius strip in the momentum space, revealing topological band crossings
protected by a nonsymmorphic symmetry. We demonstrate that breaking this
symmetry induces a topological transition manifested as gap opening at band
crossings, and further manipulate the band structure and BEC's transport by
controlling the axial synthetic magnetic flux. Our work opens the door for
using atomic quantum simulators to explore intriguing topological phenomena
intrinsic in unconventional spaces.
