Question : How to show $SL(n,\mathbb R)$ diffeomorphic to $ SO(n) \times \mathbb R^{n(n+1)/2-1}$? Also, how to show $SL(n,\mathbb C)$ diffeomorphic to $ SU(n) \times \mathbb R^{n^2-1}$?
I have no idea about it. And, I just find a similar question that I guess can work here:
$GL^+ (n,\mathbb R)$ is diffeomorphic to $SO(n) \times T^+(n, \mathbb R)$ , where $T^+(n, \mathbb R)$ is the Lie group of all opper triangular real matrices with positive diagonal entries.
Simply, we can construct a diffeomorphism by QR decomposition, that is, $$A=QR \to (Q,R)$$ Then, we check that it's well-defined and smooth and construct an inverse.
However, for the origin question, I cannot construct a diffeomorphism just as above. Please help!
Hint If the QR decomposition of a matrix in $GL^+(n, \Bbb R)$ is $A = QR$, then $$\det A = \det (QR) = \det Q \det R = \det R ,$$ so if $$\Phi : GL^+(n, \Bbb R) \to SO(n) \times T^+(n, \Bbb R)$$ is the diffeomorphism defined by that decomposition, $(Q, R)$ is in the image of $SL(n, \Bbb R) \subset GL^+(n, \Bbb R)$ iff $\det R = 1$, which implies that the restriction $$\Phi\vert_{SL(n, \Bbb R)} : SL(n, \Bbb R) \to SO(n) \times (T^+(n, \Bbb R) \cap SL(n, \Bbb R) )$$ is itself a diffeomorphism.
It remains to show that $T^+(n, \Bbb R) \cap SL(n, \Bbb R)$ is diffeomorphic to $\Bbb R^{n (n + 1) / 2 - 1}$. Any matrix $(t_{ij}) \in T^+(n, \Bbb R) \cap SL(n, \Bbb R)$ is upper-triangular and has determinant $1$, and so $$t_{11} \cdots t_{nn} = \det (t_{ij}) = 1 .$$ Thus, we can express $t_{nn}$ as a function of $t_{11}, \ldots, t_{n - 1, n - 1}$.