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Structure equations of a submanifold

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

Abstract

A differentiable mapping ı of M into M′ is called an immersion if (ı *) x is injective for every point x of M. Here ı * is the usual differential map ı * : T x (M) → T ı(x)(M′). We say then that M is immersed in M′ by ı or that M is an immersed submanifold of M′. When an immersion ı is injective, it is called an embedding of M into M′. We say then that M (or the image ı(M)) is an embedded submanifold (or, simply, a submanifold) of M′. In this sense, throughout what follows, we adopt the convention that by submanifold we mean embedded submanifold. If the dimensions of M and M′ are n and n + p, respectively, the number p is called the codimension of a submanifold M. The interested reader is referred to [5] and [33] for further information and more details.

Keywords

Structure Equation Fundamental Form Normal Bundle Orthonormal Frame Normal Part 
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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