In the paper The geometry of free fall and light propagation by Ehlers and his colleagues (Gen. Relativ. Gravit. 44 no. 6, pp. 1587–1609 (2012)), when the authors introduce the differentiable structure on page 70, they claim:
It should be realized that the representation of light in special relativity by means of ordinary cones rather than by hypersurfaces of hour-glass shape depends partly on a particular choice of differential structure.
The question is: What do they mean by saying particular choice of differential structure? In what condition may the light cone be deformed and become like hour-glass as shown in this figure?
Thank you in advance for any help.