I'm a beginner with this notion. I begin to learn it with some examples of introduction. But the definition is not clear for me.
For example, $\mathbb{R}^n \cap \{(x_1,...,x_n) : x_1^2 + x_2^3 + ... + x_n^{n+1}=1 \}$ is a differential submanifold (in $\mathbb{R}^n$) ? The answer seems to be yes. I just would like to know the steps to say it's a submanifold. Maybe with a first example, the definition will appear clearer for me.
Hint : You can use the implicit function theorem, which tells you as a corollary $X = \{x \in \Bbb R^n : f(x) = 0\}$ is a submanifold when $df_x \neq 0$. This is the cleanest proof but of course you need to know the implicit function theorem.
A less overkilled approach is to use that the equality $x_1 = \pm \sqrt{1 - x_2^3 - \dots - x_n^{n+1}}$ which express your set as the union of graph of smooth functions if $x_1 \neq 0$, which gives you a parametrization of your submanifold at every point where $x_1 \neq 0$. Applying the same argument with other coordinates gives you a covering by charts.