TITLE: What is the "submersion principle?" Showing that $SL_n(R)$ is a submanifold of $GL_n(R)$.
QUESTION [3 upvotes]: 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.

REPLY [2 votes]: 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.
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