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I appologize in advance in case this is a very trivial issue and for any mistakes due to translating stuff from my German lecture notes to English ...

A subset $M \subset \mathbb{R}^n$ is defined to be a $k$-dim submanifold of $\mathbb{R}^n$ with $1 \le k \le n-1$, if each point $m \in M$ has the following property:

There is an open set $U \subset \mathbb{R}^n $ surrounding $m$ and an open subset $V \subset \mathbb{R}^n$ related by a diffeomorphism $\Phi: U \rightarrow V$ such that

$$ \begin{eqnarray} \nonumber \Phi(U \cap M) & = & V \cap(\mathbb{R}^k \times \{ 0 \}) \doteqdot W \\ & = & \{ x =(x_1, ...,x_n) \in V | x = (x_1, ... ,x_k, 0, ..., 0) \} \end{eqnarray} $$

This basically means, that $M\cap U$ locally corresponds to $\mathbb{R}^k$.

A local chart of $M$ at point $m$ is given by the pair $(W,\phi \doteqdot \Phi^{-1})$

From another source, a k-dim manifold is described to be locally $\mathbb{R}^k$ too and covered by a family of local coordinate systems.

What is the difference between a manifold and a submanifold of the same dimension?

Is it just that the sumbanifold is "embeded" into a higher-dimensional space, whereas the manifold is not? To me it seems both can be covered in a very similar way by local coordinates.

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    $\begingroup$ Yes, a manifold is not necessarily embedded inside $\Bbb R^n$, where a submanifold of $\Bbb R^n$ by definition sits inside $\Bbb R^n$. It's a slightly nontrivial theorem that any manifold (abstract; not embedded anywhere) is homeomorphic to a submanifold of $\Bbb R^n$ for some $n$, so there's not much difference. $\endgroup$ – Balarka Sen Jun 25 '16 at 15:35
  • $\begingroup$ @Balarkasen Any manifold? :) $\endgroup$ – Pedro Tamaroff Jun 25 '16 at 17:41
  • $\begingroup$ @PedroTamaroff My manifolds are all nice (I suppose you are asking me to add the Hausdorff and second countable adjectives). $\endgroup$ – Balarka Sen Jun 25 '16 at 17:45
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An abstract manifold of dimension $k$ is defined by a family of charts (local coordinate systems). In particular, such a family of charts exists for every $k$-dimensional submanifold of $\mathbb{R}^n$.

On the other hand, manifolds may be constructed independent of a realization in a Euclidean space, e.g., by surgery. There are 2-dimensional manifolds such as the Klein bottle that cannot be realized (without self-intersection) as 2-dimensional submanifold of $\mathbb{R}^3$.

The Whitney embedding theorem says that a $k$-dimensional manifold can be diffeomorphically embedded into $\mathbb{R}^{2k}$ iff its topology is Hausdorff and second countable. Thus the concept of an abstract manifold is slightly more general than a submanifold but the collection of nice examples is the same (up to diffeomorphisms).

Note that there may be multiple differentiable structure on the same manifold considered as a topological space. Examples are exotic spheres.

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    $\begingroup$ Let us also add that submanifolds of $\Bbb R^n$ must inherit some topological properties of the ambient space, such as being separable, second-countable, paracompact etc., that abstract (i.e. not embedded) manifolds need not have. $\endgroup$ – Alex M. Jun 25 '16 at 17:46

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