Submanifold

In the narrow sense, an $ n $- dimensional topological submanifold of an $ m $- dimensional topological manifold $ M $ is a subset $ N \subset M $ which is an $ n $- dimensional manifold in the induced topology. The number $ m- n $ is called the codimension of the submanifold $ N $. One most frequently encounters locally flat submanifolds, for which the identity imbedding $ i: N \rightarrow M $ is a locally flat imbedding. A subset $ N \subset M $ is a locally flat submanifold if for each point $ p \in N $ there exists a neighbourhood $ U $ of this point in $ M $ together with local coordinates $ x _ {1} \dots x _ {m} $ in it such that in terms of these coordinates $ N \cap U $ is described by the equations $ x _ {n+} 1 = \dots = x _ {m} = 0 $.

In the broad sense, an $ n $- dimensional topological submanifold of an $ m $- dimensional topological manifold $ M $ is an $ n $- dimensional manifold $ N $ which, as a set of points, is a subset of $ M $( in other words, $ N $ is a subset of $ M $ with the structure of an $ n $- dimensional manifold) and for which the identity imbedding $ i: N \rightarrow M $ is an immersion (cf. Immersion of a manifold). A submanifold in the narrow sense is a submanifold in the broad sense, while the latter is a submanifold in the narrow sense if and only if $ i $ is an imbedding in the topological sense (i.e. for each point $ p \in N $ there are arbitrarily small neighbourhoods in $ N $ that are intersections with $ N $ of certain neighbourhoods in $ M $).

A piecewise-linear, analytic or differentiable submanifold (of class $ C ^ {l} $, $ l \leq \infty $) of a piecewise-linear, analytic or differentiable manifold $ M $( of class $ C ^ {k} $, $ l \leq k \leq \infty $) in the broad sense (or, correspondingly, in the narrow sense) is a subset $ N \subset M $ having the structure of a piecewise-linear, analytic or differentiable manifold (of class $ C ^ {l} $), where $ i $ is a piecewise-linear, analytic or differentiable immersion (of class $ C ^ {l} $) (correspondingly, an imbedding). The definition of a differentiable submanifold of class $ C ^ {l} $ is suitable also for $ l = 0 $, and it coincides in that case with the definition of a topological submanifold. It is usually understood that $ l \geq 1 $.

In the analytic and differentiable cases, the submanifold is always locally flat. Therefore, the definition of an analytic (differentiable) submanifold in the narrow sense is usually formulated from the start as an analytic (differentiable) form of the definition given in 1) for a locally flat submanifold by means of local coordinates, with the additional condition that the local coordinates $ x _ {1} \dots x _ {m} $ are analytic (differentiable of class $ C ^ {l} $). If a subset $ N $ satisfies the latter definition, it is equipped in a natural way with the structure of an analytic (differentiable of class $ C ^ {l} $) manifold, and $ i $ is an imbedding in the sense of the corresponding structure.

A piecewise-linear submanifold in the narrow sense can be locally represented as a subpolyhedron in the ambient manifold and is piecewise linearly equivalent to a simplex (cf. Simplex (abstract)). It is not always locally flat (although this is so for $ m - n > 2 $); also, for such a manifold, the property of being locally flat in the topological sense does not coincide (at least directly) with the property of being locally flat in the piecewise-linear sense.

A simple modification of these definitions gives the definitions of: a submanifold with boundary; a submanifold of a manifold with boundary (in the topological situation, it is desirable to impose a restriction on the submanifold at the boundary of the ambient manifold, see ); a submanifold the various components of which may have different dimensions; a submanifold of a manifold of infinite dimension ; and a complex-analytic submanifold of a complex-analytic manifold.

The concept of a manifold in the narrow sense is a direct generalization of the concepts of a curve and a surface. A submanifold in the broad sense is used in the theory of Lie groups (where this concept was first introduced ), in differential geometry

and also in the theory of foliations (cf. Foliation).

In algebraic geometry, a submanifold (usually called a subvariety here) is a closed subset of an algebraic variety in the Zariski topology. This formalizes the idea that a subvariety is specified by algebraic equations. In addition to the transition from $ \mathbf R $ to other fields, the change in the concept of a subvariety in this case is that one allows subvarieties with singularities.

References

[1]	V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian) MR759162
[2]	S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III MR1931083 MR1532744 MR0155257 Zbl 1008.57001 Zbl 0103.15101
[3]	C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946) MR0082628 MR0015396 Zbl 0063.00842
[4]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) MR0193578 Zbl 0129.13102

Comments

Any $ C ^ {r} $- manifold $ M ^ {n} $ can be $ C ^ {r} $ imbedded in $ \mathbf R ^ {2n} $( Whitney's imbedding theorem), and so any $ C ^ {r} $- manifold can be seen as a submanifold of some $ \mathbf R ^ {m} $.

References

[a1]	M.W. Hirsch, "Differential topology" , Springer (1976) pp. Chapt. 6 MR0448362 Zbl 0356.57001
[a2]	Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, T. Fomenko, "Introduction to topology" , Kluwer (1993) (Translated from Russian) MR1450091 MR0824983 MR0591670 Zbl 0836.57001 Zbl 0834.57001 Zbl 0478.57001