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Luciana F. Martins

Publications and source records attributed to Luciana F. Martins.

8 recordsLinked to original sources
arXiv2026

Geometry on principal curvature surfaces

We study the geometry of the principal curvature surface associated with a Whitney umbrella. This surface is obtained by extending the Whitney umbrella in the normal direction by its bounded principal curvature and admits a natural geometric interpretation. We prove that its intersection with the normal plane coincides with the pedal curve of the curvature parabola and deduce that the focal conic is the inversion of this pedal curve. We then investigate the geometry of the principal curvature surface along the exceptional set by studying its geodesic and normal curvatures, as well as its Gaussian and mean curvatures, and determining all possible generic numbers of zeros of these curvature functions.

math.DG↗
arXiv2025

Helicoidal surfaces of frontals in Euclidian space as deformations of surfaces of revolution, with singularities

We investigate helicoidal surfaces in three-dimensional Euclidean space whose profile curves are frontals. Using the framework of Legendre curves and framed surfaces, we establish conditions under which helicoidal surfaces generated by frontals are themselves frontals or fronts. We then derive curvature expressions in terms of the invariants of the generating Legendre curve. Our study extends classical results on parallel and focal surfaces of surfaces of revolution to the helicoidal setting. In particular, we show that both parallel and focal surfaces of a helicoidal surface are helicoidal, with their generating curves arising from one-parameter deformations of the corresponding parallel and evolute curves. We prove that singularities of these curves persist under such deformations, revealing geometric rigidity and stability of singularities. Finally, we examine the behavior of the Gaussian and mean curvatures near singular points of helicoidal surfaces

math.DG↗
arXiv2024

Boundedness of geometric invariants near a singularity which is a suspension of a singular curve

Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study boundedness and orders of several geometric invariants near a singular point of a surface which is a suspension of a singular curve in the plane and those of curves passing through the singular point. We evaluates the orders of Gaussian and mean curvatures and them of geodesic, normal curvatures and geodesic torsion for the curve.

math.DG↗
arXiv2016

Geometry of cuspidal edges with boundary

We study differential geometric properties of cuspidal edges with boundary. There are several differential geometric invariants which are related with the behavior of the boundary in addition to usual differential geometric invariants of cuspidal edges. We study the relation of these invariants with several other invariants.

math.DG↗
arXiv2015

Behavior of Gaussian curvature and mean curvature near non-degenerate singular points on wave fronts

We define cuspidal curvature $κ_c$ (resp. normalized cuspidal curvature $μ_c$) along cuspidal edges (resp. at swallowtail singularity) in Riemannian $3$-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product $κ_Π$ called the product curvature (resp. $μ_Π$ called normalized product curvature) of $κ_c$ (resp. $μ_c$) and the limiting normal curvature $κ_ν$ is an intrinsic invariant of the surface, and is closely related to the boundedness of the Gaussian curvature. We also consider the limiting behavior of $κ_Π$ when cuspidal edges accumulate to other singularities. Moreover, several new geometric invariants of cuspidal edges and swallowtails are given.

math.DG↗
arXiv2014

Geometric invariants of cuspidal edges

We give a normal form of the cuspidal edge which uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges which determine them up to order three. We also clarify relations between these invariants.

math.DG↗