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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2404.09963 |
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| _version_ | 1866910411060674560 |
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| author | Deolindo-Silva, Jorge Luiz |
| author_facet | Deolindo-Silva, Jorge Luiz |
| contents | A smooth ruled surface in 4-space has only parabolic points or inflection points of real type. We show, by means of contact with transverse planes, that at a parabolic point, there exist two tangent directions determining two planes along which the parallel projection exhibits $\mathcal A$-singularities of type butterfly or worse. In particular, such parabolic point can be classified as butterfly hyperbolic, parabolic, or elliptic point depending on the value of the discriminant of a binary differential equation (BDE). Also, whenever such discriminant is positive, we ensure that the integral curves of these directions form a pair of foliations on the ruled surface. Moreover, the set of points that nullify the discriminant is a regular curve transverse to the regular curve formed by inflection points of real type. Finally, using a particular projective transformation, we obtain a simple parametrization of the ruled surface such that the moduli of its 5-jet identify a butterfly hyperbolic/parabolic/elliptic point, as well as we get the stable configurations of the solutions of BDE in the discriminant curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09963 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the differential geometry of smooth ruled surfaces in 4-space Deolindo-Silva, Jorge Luiz Differential Geometry A smooth ruled surface in 4-space has only parabolic points or inflection points of real type. We show, by means of contact with transverse planes, that at a parabolic point, there exist two tangent directions determining two planes along which the parallel projection exhibits $\mathcal A$-singularities of type butterfly or worse. In particular, such parabolic point can be classified as butterfly hyperbolic, parabolic, or elliptic point depending on the value of the discriminant of a binary differential equation (BDE). Also, whenever such discriminant is positive, we ensure that the integral curves of these directions form a pair of foliations on the ruled surface. Moreover, the set of points that nullify the discriminant is a regular curve transverse to the regular curve formed by inflection points of real type. Finally, using a particular projective transformation, we obtain a simple parametrization of the ruled surface such that the moduli of its 5-jet identify a butterfly hyperbolic/parabolic/elliptic point, as well as we get the stable configurations of the solutions of BDE in the discriminant curve. |
| title | On the differential geometry of smooth ruled surfaces in 4-space |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2404.09963 |