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Envelopes of families of framed surfaces and singular solutions of first-order partial differential equations

Part of: Classical differential geometry General first-order equations and systems Differential topology Theory of singularities and catastrophe theory

Published online by Cambridge University Press:  15 September 2020

Masatomo Takahashi  and
Haiou Yu
Masatomo Takahashi
Affiliation:
Muroran Institute of Technology, Muroran050-8585, Japan (masatomo@mmm.muroran-it.ac.jp)
Haiou Yu
Affiliation:
Jilin University of Finance and Economic, Changchun130117, China (yuhaiou@jlufe.edu.cn)
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Abstract

In order to investigate envelopes for singular surfaces, we introduce one- and two-parameter families of framed surfaces and the basic invariants, respectively. By using the basic invariants, the existence and uniqueness theorems of one- and two-parameter families of framed surfaces are given. Then we define envelopes of one- and two-parameter families of framed surfaces and give the existence conditions of envelopes which are called envelope theorems. As an application of the envelope theorems, we show that the projections of singular solutions of completely integrable first-order partial differential equations are envelopes.

Keywords

Envelopefamily of framed surfacessingular solution

MSC classification

Primary: 58K05: Critical points of functions and mappings
Secondary: 57R45: Singularities of differentiable mappings 53A05: Surfaces in Euclidean space 53A55: Differential invariants (local theory), geometric objects 35F25: Initial value problems for nonlinear first-order equations 35F55: Initial value problems for nonlinear first-order systems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 5 , October 2021 , pp. 1515 - 1542
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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