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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.

Moishe Kohan
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Mélanie De la Cheminée
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  • You need to check: 1. That it is a topological space which is second countable and Haudsdorff (which is obviously true viewing it as subspace of $\mathbb{R}^n$). 2. That it has an atlas - consider stereographic projection. 3. That inclusion map into $\mathbb{R}^n$ is an embedding (which obviously is due to its subspace topology) – user160738 Jun 13 '17 at 17:16
  • @user160738 what is the stereographic projection ? This is not the sphere. –  Jun 13 '17 at 17:26
  • This is not the sphere. The equation here is $x_1^2 + x_2^3 + \dots + x_n^{n+1} = 1$, which is unbounded so surely not diffeomorphic to the sphere. –  Jun 13 '17 at 17:34
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    @N.H. Sorry, you're right, I thought all superscripts were $2$ – user160738 Jun 13 '17 at 17:36

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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.