Suppose now I have two smooth manifolds of dimension $n-1$, which are given by the zero level sets of two polynomials. Specifically, suppose $M$ is the manifold given by the zero set of the polynomial $p_1: \mathbb{R}^n\to\mathbb{R}$ and $N$ is the manifold given by the zero set of the polynomial $p_2: \mathbb{R}^n\to\mathbb{R}$. $n\ge 2$. Suppose $M$ and $N$ are different and each of them only have one connected component. I am wondering is it possible that the two manifolds intersect at a submanifold of dimension also $n-1$?
Probably the statement of the question is not hundred percent correct, since I think the intersection might not be a submanifold. But what I want to ask is that is it possible that two curves defined by the zero level set of two polynomials share a common part(sub-arc)? Is it possible that two surfaces given by the zero level sets of two polynomials share a common part (a sub-surface)? And also the cases for higher dimensional manifolds. Again, the zero level set do not contain any critical points.
Note: By regular value theorem, we have that to let the zero level set be a manifold, we require that the level set do not contain any critical point. So, $M$ and $N$ do not have any critical point.
Best Answer
Yes, it is very possible. Consider, for example, the circle $x^2+y^2=1$ and the parabola $y=x^2-\frac54$. You can make higher-dimensional examples quite easily.
EDIT (responding to the totally different question): With polynomials or real analytic functions, this cannot happen unless the hypersurfaces coincide (at least on a connected component). However, in the smooth category, you can use bump functions to create two hypersurfaces that coincide on a large (closed) subset (with interior) and then diverge. Note that they will coincide on a manifold with boundary. If you require that the set of coincidence have no boundary, then they'll have to agree on a whole connected component, of course.