Surface theory and low-dimensional topology by using singularity theory

2014 Fiscal Year Final Research Report

• Project/Area Number: 23740045
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Kobe University (2012-2013); Gifu University (2011)
• Principal Investigator: SAJI Kentaro 神戸大学, 理学(系)研究科(研究院), 准教授 (70451432)
• Project Period (FY): 2011 – 2014
• Keywords: 特異点 / 特異点の認識問題 / カスプ辺 / ガウス写像

Outline of Final Research Achievements

This research was forcused on two things.
First one is criteria for singularities. About this, we got a useful criteria for Morin singulariteis where the dimensions of source and target are different. We made a mathod to construct functions which characterize the singular set of given map, and using it, we got conditions for Morin singularities. Moreover, criteria for rhamphid cusp is obtained.
Second one is to investigate surfaces with singularity using criteria for singularities. We construct a notion ``coherent tangent bundle'', giving an intrinsic formulation for wave fronts. We show Gauss-Bonnet type theorem for it and give several other applications. Furthermore, using a duality between surfaces in 3-sphere, we study conditions for cuspidal edge and swallowtail of surface whose Gauss map is a curve. Moreover, we show duality of conditions of singularities.

Free Research Field

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