I was reading this post that explains why
$$\phi :O_{n}\times H\rightarrow GL_{n}(\mathbb{R})$$ $$\phi(B,A)=BA$$ is a diffeomorphism. Here $O_{n}$ is the orthogonal group and $H$ is the group of all upper triangular matices.
The answer uses the following theorem.
Theorem 7.25 (Equivariant Rank Theorem). Let $M$ and $N$ be smooth manifolds and let $G$ be a Lie group. Suppose $F : M \to N$ is a smooth map that is equivariant with respect to a transitive smooth $G$-action on $M$ and any smooth $G$-action on $N$ . Then $F$ has constant rank. Thus, if $F$ is surjective, it is a smooth submersion; if it is injective, it is a smooth immersion; and if it is bijective, it is a diffeomorphism.
but in the answer, he just finds one action on $O_n \times H$ and another one on $GL_n(\mathbb{R})$ such that the map is equivariant to conclude. But the theorem says that to conclude the map should be equivariant for any action on $GL_n(\mathbb{R})$ no? What am I understanding wrong here?
This quotation is from my Introduction to Smooth Manifolds. It's a good illustration of the pitfalls of using the word "any" as a quantifier.
What I meant by "any smooth $G$-action on $N$" was "an arbitrary (not necessarily transitive) smooth $G$-action on $N$." I definitely did not mean "every smooth $G$-action on $N$." I expect that except in very trivial cases, there won't be any maps that are equivariant with respect to a transitive action on $M$ and every action on $N$.