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 structure is inherited from the ambient space is not generally agreed upon. So let's start with differentiable submanifolds of ${\mathbb{R}}^n$ as that's the most useful case.

If $\rho_k$ are in fact smooth then $M$ is a smooth submanifold and similarly if $\rho$ is real analytic then $M$ is a real analytic submanifold. If we identify ${\mathbb{R}}^{2n}$ with ${\mathbb{C}}^n$ and we have a submanifold there it is called a real submanifold in ${\mathbb{C}}^n$ $\rho_k$ 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 $N \subset M$ is said to have the submanifold property if there exists an integer $n \leq m$ such that for each $p \in N$ there is a coordinate neighbourhood $U$ and a coordinate function $\varphi \colon U \to {\mathbb{R}}^m$ of $M$ such that $\varphi(p) = (0,0,0,\ldots,0)$ $\varphi(U \cap N) = \{ x \in \varphi(U) \mid x_{n+1} = x_{n+2} = \ldots = x_m = 0 \}$ if $n < m$ or $N \cap U = U$ if $n=m$

The ambiguity arises about what topology 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 union 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 subspace 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 map $i \colon N \to M$ is an imbedding, then we say that $N$ is an imbedded (or embedded) submanifold of $M$

If $N \subset M$ 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 regular and smooth submanifold in fact) is the unit sphere in ${\mathbb{R}}^n$ This is in fact a hypersurface as it is of codimension 1.

Bibliography

1

William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.

2

M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.