Transversal intersection of the hypersurfaces with the sphere

Question

Let $X=f^{-1}(y)$ be a n-1-dim smooth manifold in $\mathbb{R}^n$. Assume that $X$ intersect transversaly a sphere $S_{n-1}\subset \mathbb{R}^n$ (i.e. $f|_{S_{n-1}}$ don't have a singular point). Is it true that $X\cap S_{n-1}$ is difeomorphic to the sum of spheres $S_{n-2}\subset \mathbb{R}^n$?

Answer 1

No, it's not true. Here's an example for you to work out:

Define $f\colon\Bbb R^4-\{0\}\to\Bbb R$ by $f(x,y,z,w) = xy-wz$, and let $X = f^{-1}(0)\subset\Bbb R^4-\{0\}$. Then $X\cap S^3$ is diffeomorphic to a torus, i.e., $S^1\times S^1$. (If you insist on having the domain be all of $\Bbb R^4$, I can use a bump function to modify $f$ in a small neighborhood of the origin without affecting what happens along $S^3$.)

You can do something analogous starting in $\Bbb R^6$ and ending up with $S^2\times S^2$. This is particularly interesting because it is (the Plücker embedding of) the Grassmannian of oriented $2$-planes in $\Bbb R^4$.