Is $S$ a regular submanifold?

Question

$M=M_{n\times n}(\Bbb R)$

$S=\operatorname{SL}(n, \Bbb R) = \left \{ A \in M \mid \det(A)=1 \right \}$

$M$ is an $n^{2}$ dimensional $C^{\infty}$ manifold.

Is $S$ a regular submanifold?

Answer 1

Yes, the determinant function $\det:M_{n\times n}(\mathbb{R})\to\mathbb{R}$ is smooth and has constant rank, then $\det^{-1}(1)=\operatorname{sl}(n,\mathbb{R})$ is a regular submanifold. Please see Boothby's book, page 79, theorem 5.8.