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Codimension reduction of a submanifold

  • Mirjana Djorić
  • Masafumi Okumura
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 19)

Abstract

Let us first recall the theory of curves in 3-dimensional Euclidean space E 3. The curve C, whose torsion vanishes identically, is a plane curve. In other words, for the curve C without torsion, there exists a 2-dimensional totally geodesic subspace E 2 such that \(C\subset {\bf E}^2\subset {\bf E}^3\). In general, a curve C is a submanifold of codimension 2 of E 3, but if its torsion is zero, it can be regarded as a submanifold of codimension 1 in E 2, that is, the codimension is reduced from 2 to 1.

Keywords

Normal Space Orthogonal Complement Complex Projective Space Constant Dimension Parallel Translation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.University of BelgradeBelgradeSerbia
  2. 2.Saitama UniversitySaitamaJapan

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