Codimension reduction of a submanifold

Part of the Developments in Mathematics book series (DEVM, volume 19)


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.


Normal Space Orthogonal Complement Complex Projective Space Constant Dimension Parallel Translation 


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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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