In this paper, we study concircular surfaces, defined as surfaces whose unit normal vector field $ \mathcal{N} $ and a given concircular vector field $ \mathcal{Y} $ satisfy $ \langle\mathcal{N}, \mathcal{Y}\rangle = $ constant. Applying singularity theory, we provide a classification of singularities on such surfaces, including cuspidal edges and swallowtails. We find that the order of contact between the curve $ (\boldsymbol{\alpha }, \mathcal{N}) $ associated with the concircular surface and the affine tangent bundle of the unit sphere is closely related to these singularities. Furthermore, from the perspective of spherical Legendrian duality, we establish a direct relationship between the rulings and the normal vector fields on the surface. Finally, several examples are presented to illustrate the main conclusions.
Citation: Jie Huang. Singularities of concircular surfaces in Euclidean 3-space[J]. AIMS Mathematics, 2026, 11(8): 26246-26259. doi: 10.3934/math.20261052
In this paper, we study concircular surfaces, defined as surfaces whose unit normal vector field $ \mathcal{N} $ and a given concircular vector field $ \mathcal{Y} $ satisfy $ \langle\mathcal{N}, \mathcal{Y}\rangle = $ constant. Applying singularity theory, we provide a classification of singularities on such surfaces, including cuspidal edges and swallowtails. We find that the order of contact between the curve $ (\boldsymbol{\alpha }, \mathcal{N}) $ associated with the concircular surface and the affine tangent bundle of the unit sphere is closely related to these singularities. Furthermore, from the perspective of spherical Legendrian duality, we establish a direct relationship between the rulings and the normal vector fields on the surface. Finally, several examples are presented to illustrate the main conclusions.
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Jie Huang. Singularities of concircular surfaces in Euclidean 3-space[J]. AIMS Mathematics, 2026, 11(8): 26246-26259. doi: 10.3934/math.20261052
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