2024/06/10 by Saji, Kentaro, Umehara, Masaaki, Yamada, Kotaro
#53A10 #53A35 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2406.05971
This is a continuation of the authors' earlier work on deformations of cuspidal edges. We give a representation formula for swallowtails in the Euclidean 3-space. Using this, we investigate map germs of generic swallowtails in 3-dimensional space from, and show some important properties of them. In particular, we give a representation formula giving all map germs of swallowtails in the Euclidean 3-space whose Gaussian curvatures are bounded from below by a positive constant or by a negative constant from above. Using this, we show that any swallowtails are deformed into a swallowtail of constant Gaussian curvature preserving the sign of their Gaussian curvatures.