There are several conflicting definitions of what a submanifold is, depending on which author you are reading. All that agrees is that a submanifold is a subset of a manifold which is itself a manifold, however how structureMathworldPlanetmath is inherited from the ambient space is not generally agreed upon. So let’s start with differentiableMathworldPlanetmathPlanetmath submanifolds of n as that’s the most useful case.


Let M be a subset of n such that for every point pM there exists a neighbourhood Up of p in n and m continuously differentiable functions ρk:U where the differentials of ρk are linearly independentMathworldPlanetmath, such that


Then M is called a submanifold of n of dimensionMathworldPlanetmathPlanetmathPlanetmath m and of codimension n-m.

If ρk are in fact smooth then M is a smooth submanifold and similarly if ρ is real analytic then M is a real analytic submanifold. If we identify 2n with n and we have a submanifold there it is called a real submanifold in n. ρk are usually called the local defining functions.

Let’s now look at a more general definition. Let M be a manifold of dimension m. A subset NM is said to have the submanifold property if there exists an integer nm, such that for each pN there is a coordinatePlanetmathPlanetmath neighbourhood U and a coordinate function φ:Um of M such that φ(p)=(0,0,0,,0), φ(UN)={xφ(U)xn+1=xn+2==xm=0} if n<m or NU=U if n=m.


Let M be a manifold of dimension m. A subset NM with the submanifold property for some nm is called a submanifold of M of dimension n and of codimension m-n.

The ambiguity arises about what topologyMathworldPlanetmath we require N to have. Some authors require N to have the relative topology inherited from M, others don’t.

One could also mean that a subset is a submanifold if it is a disjoint unionMathworldPlanetmath of submanifolds of different dimensions. It is not hard to see that if N is connected this is not an issue (whatever the topology on N is).

In case of differentiable manifolds, if we take N to be a subspaceMathworldPlanetmath of M (the topology on N is the relative topology inherited from M) and the differentiable structure of N to be the one determined by the coordinate neighbourhoods above then we call N a regular submanifold.

If N is a submanifold and the inclusion mapMathworldPlanetmath i:NM is an imbedding, then we say that N is an imbedded (or embedded) submanifold of M.


Let pM where M is a manifold. Then the equivalence classMathworldPlanetmathPlanetmath of all submanifolds NM such that pN where we say N1 is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to N2 if there is some open neighbourhood U of p such that N1U=N2U is called the germ of a submanifold through the point p.

If NM is an open subset of M, then N is called the open submanifold of M. This is the easiest class of examples of submanifolds.

Example of a submanifold (a in fact) is the unit sphere in n. This is in fact a hypersurface as it is of codimension 1.


  • 1 William M. Boothby. , Academic Press, San Diego, California, 2003.
  • 2 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
Title submanifold
Canonical name Submanifold
Date of creation 2013-03-22 14:47:20
Last modified on 2013-03-22 14:47:20
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 8
Author jirka (4157)
Entry type Definition
Classification msc 32V40
Classification msc 53C40
Classification msc 53B25
Classification msc 57N99
Related topic Manifold
Related topic Hypersurface
Defines real submanifold
Defines codimension of a manifold
Defines local defining functions
Defines real submanifold
Defines smooth submanifold
Defines real analytic submanifold
Defines regular submanifold
Defines imbedded submanifold
Defines embedded submanifold
Defines germ of a submanifold
Defines open submanifold