View source for Submanifold

In the narrow sense, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s0909201.png" />-dimensional topological submanifold of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s0909202.png" />-dimensional topological [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s0909203.png" /> is a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s0909204.png" /> which is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s0909205.png" />-dimensional manifold in the induced topology. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s0909206.png" /> is called the codimension of the submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s0909207.png" />. One most frequently encounters locally flat submanifolds, for which the identity imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s0909208.png" /> is a [[Locally flat imbedding|locally flat imbedding]]. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s0909209.png" /> is a locally flat submanifold if for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092010.png" /> there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092011.png" /> of this point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092012.png" /> together with local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092013.png" /> in it such that in terms of these coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092014.png" /> is described by the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092015.png" />.

In the broad sense, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092016.png" />-dimensional topological submanifold of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092017.png" />-dimensional topological manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092018.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092019.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092020.png" /> which, as a set of points, is a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092021.png" /> (in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092022.png" /> is a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092023.png" /> with the structure of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092024.png" />-dimensional manifold) and for which the identity imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092025.png" /> is an immersion (cf. [[Immersion of a manifold|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092026.png" /> is an imbedding in the topological sense (i.e. for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092027.png" /> there are arbitrarily small neighbourhoods in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092028.png" /> that are intersections with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092029.png" /> of certain neighbourhoods in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092030.png" />).

A piecewise-linear, analytic or differentiable submanifold (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092032.png" />) of a piecewise-linear, analytic or differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092033.png" /> (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092035.png" />) in the broad sense (or, correspondingly, in the narrow sense) is a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092036.png" /> having the structure of a piecewise-linear, analytic or differentiable manifold (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092037.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092038.png" /> is a piecewise-linear, analytic or differentiable immersion (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092039.png" />) (correspondingly, an imbedding). The definition of a differentiable submanifold of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092040.png" /> is suitable also for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092041.png" />, and it coincides in that case with the definition of a topological submanifold. It is usually understood that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092042.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092043.png" /> are analytic (differentiable of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092044.png" />). If a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092045.png" /> satisfies the latter definition, it is equipped in a natural way with the structure of an analytic (differentiable of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092046.png" />) manifold, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092047.png" /> 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)|Simplex (abstract)]]). It is not always locally flat (although this is so for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092048.png" />); 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|Foliation]]).

In [[Algebraic geometry|algebraic geometry]], a submanifold (usually called a subvariety here) is a closed subset of an algebraic variety in the [[Zariski topology|Zariski topology]]. This formalizes the idea that a subvariety is specified by algebraic equations. In addition to the transition from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092049.png" /> to other fields, the change in the concept of a subvariety in this case is that one allows subvarieties with singularities.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Rokhlin,   D.B. Fuks,   "Beginner's course in topology. Geometric chapters" , Springer  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,   "Introduction to differentiable manifolds" , Interscience  (1967)  pp. App. III</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Chevalley,   "Theory of Lie groups" , '''1''' , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Sternberg,   "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR></table>



====Comments====
Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092050.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092051.png" /> can be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092052.png" /> imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092053.png" /> (Whitney's imbedding theorem), and so any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092054.png" />-manifold can be seen as a submanifold of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090920/s09092055.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.W. Hirsch,   "Differential topology" , Springer  (1976)  pp. Chapt. 6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Yu. Borisovich,   N. Bliznyakov,   Ya. Izrailevich,   T. Fomenko,   "Introduction to topology" , Kluwer  (1993)  (Translated from Russian)</TD></TR></table>