Regular Values and Level Sets

Let \(\omega =dx\wedge dy\) be the 2-form on \(\mathbb {R}^3\). Let \(h:\mathbb {R}^3\to \mathbb {R}\) be smooth so that 0 is a regular value. Let \(Z=h^{-1}(0)\). Assume that \(\frac {\partial h}{\partial z}\ne 0\) at all points of \(Z\). Show that the restriction of \(\omega \) to \(Z\) is a volume form on \(Z\).

Let \(\mathcal {M}(n)\) denote the space of all \(n\times n\) matrices with real entries, iden- tified with \(\mathbb {R}^{n^2}\), and let \(S(n)\) be the subspace of symmetric matrices. Consider the smooth map \(F:\mathcal {M}(n)\to S(n)\) defined by \(f(A)=AA^{\top }\), where \(A^{\top }\) is the transpose of \(A\).

1. Show that the identity matrix \(I\) is a regular value of \(f\) .

2. Deduce that the orthogonal group

   \[ O(n)=\{A\in \mathcal {M}(n)\mid AA^{\top }=I\} \]

   is a submanifold of \(\mathcal {M}(n)\). Carefully state any theorems you are citing.

1. State the definition of a regular value and state the pre-image theorem.
2. Let \(F:\mathbb {R}^n\to \mathbb {R}^m\) be a smooth function with \(y\in \mathbb {R}^m\) a regular value and let \(M=F^{-1}(y)\). If \(f:\mathbb {R}^n\to \mathbb {R}^k\) is a smooth function and \(x\in M\) is critical point for \(f|_M\) what are the possible values for the dimension of \(\ker F_*(x)\cap \ker f_*(x)\)? (Here \(F_*(x)\) is the tangent map from \(T_x\mathbb {R}^n\to T_y\mathbb {R}^m\) and \(f_*(x)\) is also a tangent map.)

3. Define \(G:\mathbb {R}^n\to \mathbb {R}^{m+k}\) by \(G(x)=(F(x),f(x))\).

   1. If \(z\in \mathbb {R}^k\) is regular value of \(f|_M\) is \((y,z)\in \mathbb {R}^{m+k}\) always a regular value of \(G\)? Give a proof or find a counterexample.
   2. If \(z\in \mathbb {R}^k\) is a regular of \(f\) (as a function on \(\mathbb {R}^n\)) is \((y,z)\in \mathbb {R}^{m+k}\) always a regular value of \(G\). Give a proof or find a counterexample.

Let \(S^2=\{(x,y,z)\in \mathbb {R}^3\mid x^2+y^2+z^2=1\}\) be the two sphere in \(\mathbb {R}^3\). Define \(f:\mathbb {R}^3\to \mathbb {R}\) by \(f(x,y,z)=x(y-2)\).

1. Show that 0 is a critical value of \(f\) and find all critical points in \(\mathbb {R}^3\).
2. Show that 0 is a regular value of \(f\) restricted to \(S^2\) and conclude that \(M=f^{-1}(0)\cap S^2\) is manifold. What manifold is \(M\)?
3. Define \(g:\mathbb {R}^3\to \mathbb {R}\) by \(g(x,y,z)=x+z\). Calculate the critical points of \(g\) restricted to \(S^2\) and the critical points of \(g\) restricted to \(M\).

Let \(f:M\to N\) be a differentiable map between \(n\)-manifolds and assume that \(M\) is compact and \(N\) is connected. If \(x\in N\) is a regular value of \(f\) show that \(f^{-1}(x)\) is finitely many points in \(M\). If every point in \(N\) is a regular value of \(f\) define a function \(\#:N\to \mathbb {Z}\) by letting \(\#(x)\) be the number of points in \(f^{-1}(x)\) and show that \(\#\) is constant.

Let \(f:M\to N\) be a smooth map between smooth manifolds, and let \(y\) be a regular value of this map. If \(M\) and \(N\) are compact, oriented, and have the same dimension, then what is the definition of \(\deg (f;y)\), the degree of \(f\) measured at the value \(y\)? We have seen that the degree is independent of the regular value chosen, and an ingredient of that proof is the following lemma:

Suppose \(Q\) is a compact smooth manifold and that \(\partial Q=M\). If \(F:Q\to N\) is smooth, and \(z\) is a regular value of \(F|_{\partial Q}\), then \(\deg (F|_{\partial Q};z)=0\).

Prove the lemma.

1. What theorem from multivariable calculus is the proof of the regular value theorem dependent on?
2. What theorem from single variable calculus is the proof of Stokes’ theorem dependent on?
3. What theorem from differential topology is used to prove that transverse intersection of manifolds is a manifold?

Let \(M\) and \(N\) be smooth, closed, connected \(n\)-manifolds, and let \(f:M\to N\) be a smooth map. If every point of \(N\) is a regular value, show that the cardinality of the pre-image of a point in \(N\) defines a constant function on \(N\).

Let \(f:\Sigma \to S^1\) be a smooth map from the surface of genus 2 to the circle. Suppose that \(y\in S^1\) is a regular value of \(f\) and that \(f^{-1}(y)\) is a nonseparating circle in \(\Sigma \) (i.e. \(\Sigma \setminus f^{-1}(y)\) is connected). Prove that \(f\) is not nullhomotopic.

Let \(p(z)=a_nz^n+\cdots +a_0\) be a non-constant complex polynomial. Prove \(p:\mathbb {C}\to \mathbb {C}\) is proper, has only a finite number of singular values, that the inverse images of regular values have constant cardinality. Deduce the Fundamental Theorem of Algebra: \(p\) has a root.