Framed Surfaces in the Euclidean Space

Abstract

Bulletin of the Brazilian Mathematical Society manuscript No.

Singularities of Non-Developable Surfaces in Three-Dimensional Euclidean Space

“…In this section, we study the ruled surface generated by Frenet-type framed base curves (see [15][16][17]). Since there exist singular points on these surfaces, in general, we cannot construct the normal vector of these surfaces.…”

Surfaces of Revolution of Frontals in the Euclidean Space

“…Theorem 2.5 (Existence Theorem for framed surfaces [10]) Let U be a simply connected domain in R 2 and let a i , b i , e i , f i , g i : U → R, i = 1, 2 be smooth functions with the integrability conditions (2.4). Then there exists a framed surface (x, n, s) : U → R 3 × ∆ whose associated basic invariants is a i , b i , e i , f i , g i , i = 1, 2.…”

“…Theorem 2.6 (Uniqueness Theorem for framed surfaces [10]) Let (x, n, s), ( x, n, s) : U → R 3 × ∆ be framed surfaces with basic invariants (G, F 1 , F 2 ) and ( G, F 1 , F 2 ), respectively. Then (x, n, s) and ( x, n, s) are congruent as framed surfaces if and only if (G, F 1 , F 2 ) and ( G, F 1 , F 2 ) coincide.…”

“…[10]) Let (x, n, s) : U → R 3 × ∆ be a framed surface and p ∈ U. (1) x is an immersion (a regular surface) around p if and only if J F (p) = 0.…”

Geometric equivalence among smooth map germs

“…Recently, R × SO(3)-equivalence has been used for the study of differential geometry of singular surfaces in R 3 (cf. [13,15,24,25,38]).…”