Differential geometry of invariant surfaces in simply isotropic and pseudo-isotropic spaces

Luiz C. B. da Silva (Lead / Corresponding author)

Research output: Contribution to journalArticlepeer-review

Abstract

We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions. Basically, the simply and pseudo isotropic geometries are the study of a three-dimensional space equipped with a rank 2 metric of index zero and one, respectively. We show that the one-parameter subgroups of isotropic rigid motions lead to seven types of invariant surfaces, which then generalizes the study of revolution and helicoidal surfaces in Euclidean and Lorentzian spaces to the context of singular metrics. After computing the two fundamental forms of these surfaces and their Gaussian and mean curvatures, we solve the corresponding problem of prescribed curvature for invariant surfaces whose generating curves lie on a plane containing the degenerated direction.
Original languageEnglish
Pages (from-to)15-52
Number of pages38
JournalMathematical Journal of the Okayama University
Volume63
Issue number1
DOIs
Publication statusPublished - Jan 2021

Keywords

  • Simply isotropic space
  • Pseudo-isotropic space
  • Singular metric
  • Invariant surface
  • Prescribed Gaussian curvature
  • Prescribed mean curvature

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