The Question is: Let $W_c = \{ ( x,y,z,w) \in R^4 | xyz = c \}$ and $Y_c = \{ ( x,y,z,w) \in R^4 | xzw = c \}$. For what real numbers $c$ is $Y_c$ a three-manifold? For what pairs $(c1,c2)$ is $W_{c_1} \cap Y_{c_2}$ a two-manifold?
I think that we can say for the first half that $Y_c$ is a three-manifold when $c \neq 0$. This isn't exactly fleshed out, but I can imagine what $xyz = c$ looks like for $c \neq 0$ with none of the variables being $0$, and locally this basically is a piece of $\mathbb{R}^3$. But my intuition is really lacking on the intersection of manifolds, and I want to say something like when both $c_1$ and $c_1$ are non-zero, as their $x$ and $z$ variables are the ones we need to intersect to form a manifold. Is this the right way of thinking about this problem, and what gaps could I fill in to explain this more concisely? Thanks for your help!
You need to make sure that $(c_1,c_2)$ is a regular value of $f\colon\mathbb R^4\to\mathbb R^2$ given by $f(x,y,z,w)=(xyz,xzw)$. That is, you need to check that $\text{rank}(Df)=2$ at every point $(x,y,z,w)$ with $f(x,y,z,w)=(c_1,c_2)$. This is equivalent to checking that the gradient vectors of the components are everywhere non-parallel on the level set.