Structure equations of a submanifold
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  and  for further information and more details.
KeywordsStructure Equation Fundamental Form Normal Bundle Orthonormal Frame Normal Part
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