On the cobordisms of M\"obius circles

The boundary of a M"obius manifold carries a canonical M"obius structure. This enables one to define the cobordism group of $n$-dimensional (closed) M"obius manifolds. The purpose of this note is to show that the cobordism group of M"obius circles is zero, i.e., every M"obius circle bounds a M"obius surface. We also complete N. Kuiper’s classification of projective structures on $S^1$ (we show that there are in fact two series of projective circles with parabolic holonomy, and not one).