# What is the “submersion principle?” Showing that $SL_n(R)$ is a submanifold of $GL_n(R)$.

I'm watching the following series of video lectures on Lie groups. In the last couple of minutes of the first lecture, he states his strategy to show that $\textrm{SL}_n(\mathbb{R})$ is a Lie subgroup of $\textrm{GL}_n(\mathbb{R})$:

1 . Show that $\textrm{SL}_n(\mathbb{R}) = \textrm{det}^{-1}\{1\}$ is a submanifold of $\textrm{GL}_n(\mathbb{R})$ at $I_n$, by showing that the map on tangent spaces at $I_n$ is surjective, and then using the "submersion principle."

2 . Use homogeneity to show that $\textrm{SL}_n(\mathbb{R})$ is a submanifold everywhere.

The second principle is clear to me, as well as the fact that the tangent space map at the identity is surjective. But I don't understand what is the "submersion principle" or how it is used. I tried googling submersion principle but nothing useful came up. This seems to have something to do with a smooth map having constant rank in a neighborhood of a point.

Edit: Not a duplicate of the previous question of why SL is a submanifold, because I am asking about a specific approach to showing it is a submanifold.

You're right. By note that $SL_n(R) = det^{-1}(1)$ with $det : GL_n(\mathbb{R}) \rightarrow \mathbb{R}\smallsetminus\{0\}$ is a constant rank map, $SL_n(R)$ is a embedded submanifold of $GL_n(R)$. The determinant function above is a constant rank map because its a smooth map which is also a group homomorphism. By a little work we can show every smooth map between Lie group that is also group homomorphism (called Lie group homomorphism) has constant rank.
• But where does the structure of a manifold on $\textrm{SL}_n(\mathbb{R})$ come from? What general principle are you using to conclude that? – D_S Nov 25 '17 at 2:01
• The structure of SL comes from the fact that its a level set of identity in $R\smallsetminus 0$. We can conclude that by constant rank level set theorem Th.5.12. – Sou Nov 25 '17 at 2:05