For any point \(p\) of a submanifold \(M\) in an Euclidean space \(\mathbb{R}^{n+r}\), it is already assigned the global coordinates in \(\mathbb{R}^{n+r}\). Here \(\mathbb{R}^{n+r}\) plays the role of the absolute space-time proposed by Newton. Then, the submanifold \(M\) is defined as: for each point \(p\) in \(M\), there exists a neighborhood \(U\), where \(r\) coordinate components can be differentiably represented by the remaining \(n\) components. Therefore, these remaining coordinate components are indepedent, the number of which is the dimension of the submanifold \(M\).

Another understanding about submanifold is not based on the above explicit representation of some \(r\) coordinate components by the \(n\) independent components, but is described as the common locus of a set of constraint functions \(F(x)=0\) or \(F(x)=t\), where 0 or \(t\) belongs to \(\mathbb{R}^{r}\) and there are \(r\) equations in the system. Then we need to check the Jacobian matrix of the map \(F(x)\). If it has rank \(r\), \(M\) is a \(n\)-dimensional submanifold in \(\mathbb{R}^{n+r}\). With the help of implicit function theorem, this definition is equivalent to the first one.

Here we should bear in mind that the rank of the Jacobian matrix is actually the number of the constraints instead of the number of free variables or dimensions of the submanifold. Hence, the submanifold dimension is the co-dimension of \(r\) in \(\mathbb{R}^{r+n}\), i.e. \(n\).

For the definition of a manifold \(M\), the global Euclidean space is not mandatory and the absolute space-time notion is abandoned. Instead, it relies on two points. Assume there is an open covering of \(M\),

  1. for each open set \(U\) in this covering, there is a one-to-one correspondence between \(U\) and an open set in \(\mathbb{R}^{n}\). N.B. At the moment, we only have this local bijection, but not a homeomorphism, since no topology has been constructed yet. In this way, each open set in the covering of \(M\) is assigned a coordinate chart.
  2. The coordinate transformation between any pair of these coordinate charts is differentiable.

The definition of a submanifold \(M^r\) in the manifold \(M^n\) is similar to that for the submanifold in an Euclidean space. But now there is no global coordinate frame any more, but only a collection of local coordinate charts on \(M^n\). For each point \(p\) in \(M^r\), it must be contained in an open set \(U\) in the open covering of \(M^n\), which is assigned a local coordinate chart. With respect to this chart, \(U \cap M^r\) can be represented as a locus by a system of constraints, the Jacobian matrix of which has rank \(n-r\). Or in another way, \(n-r\) coordinate components can be locally and differentiably represented by the remaining \(r\) coordinate components.