Intrinsic properties of surfaces with singularities

Author(s)

Hasegawa, Masaru ^[1] ; Honda, Atsufumi ^[2] ; Naokawa, Kosuke ^[3] ; Saji, Kentaro ^[3] ; Umehara, Masaaki ^[4] ; Yamada, Kotaro ^[5]

Affiliation

^[1] Univ Sao Paulo, Inst Ciencias Matemat & Comp, BR-13566590 Sao Carlos, SP - Brazil

^[2] Miyakonojo Natl Coll Technol, Miyazaki 8858567 - Japan

^[3] Kobe Univ, Dept Math, Fac Sci, Nada Ku, Kobe, Hyogo 6578501 - Japan

^[4] Tokyo Inst Technol, Dept Math & Comp Sci, Meguro Ku, Tokyo 1528552 - Japan

^[5] Tokyo Inst Technol, Dept Math, Meguro Ku, Tokyo 1528551 - Japan

• Document type: Journal article
• Source: INTERNATIONAL JOURNAL OF MATHEMATICS; v. 26, n. 4 MAR 2015.
• Web of Science Citations: 11

Abstract

In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in R-3 are Kossowski metrics, and the pull-back metrics of surfaces consisting only of cross cap singularities are Whitney metrics. Since the singular sets of Kossowski metrics are the union of regular curves on the domains of definitions, and Whitney metrics admit only isolated singularities, these two classes of metrics are disjoint. In this paper, we give several characterizations of intrinsic invariants of cuspidal edges and cross caps in these classes of metrics. Moreover, we prove Gauss-Bonnet type formulas for Kossowski metrics and for Whitney metrics on compact 2-manifolds. (AU)

• FAPESP's process: 13/02543-1 - The geometry of singular surfaces from the singularity theory viewpoint
• Grantee: Hasegawa Masaru
• Support Opportunities: Scholarships in Brazil - Post-Doctoral