Structure equations of a submanifold

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.