Prove S is a manifold.

Question

At the moment the definition of a manifold I'm working with is that of a set $X$ equipped with a smooth atlas $A$. I want to prove that $\{(a,b)\in \mathbb{R}^n\times\mathbb{R}^n \mid a\cdot a=b \cdot b=1, a\cdot b=0\}$ is a manifold. The hint I've been given is to consider the expansion of $ \ \dot a$ over an orthonormal basis $\{a,b,n_2, \dots , n_n\}$ for $\mathbb{R}^n$ but I'm not sure what's going on really. I think the aim is to prove that defining equations have constant rank by using derivatives. As you can tell I'm very confused!

Answer 1

Hint: Show that $F:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb R^3: F(a,b)=(a\cdot a,b\cdot b,a\cdot b)$ is a submersion on $\mathbb R^n\times \mathbb R^n-\{\{0\}\times \mathbb R^n\cup \mathbb R^n\times \{0\}\}$.