6510-2022-01

Let \(X\subset \mathbb {R}^3\) be a smoothly embedded torus. Show that there is a plane \(P\subset \mathbb {R}^3\) such that \(X\cap P\) is a nonempty collection of circles.

Recall that the orthogonal group \(O(n)\) is

where \(I\) is the identity matrix. Identifying the set \(\mathcal {M}_{n\times n}\) of all real \(n\times n\) matrices with \(\mathbb {R}^{n^2}\), show that \(O(n)\) is a submanifold of \(\mathcal {M}_{n\times n}\) and compute the tangent space to \(O(n)\) at \(I\) as a linear subspace of \(T_I\mathcal {M}_{n\times n}\), which is identified with \(\mathcal {M}_{n\times n}\).

In this problem you are allowed to use the fact that every closed 1-form on the 2-sphere \(S^2\) is exact and every 2-form \(\omega \) on \(S^2\) such that \(\int _{S^2}\omega =0\) is exact. You are also allowed to use that \(\int _{S^2} a^*(\omega )=-\int _{S^2}\omega \) for every 2-form \(\omega \) on \(S^2\), where \(a:S^2\to S^2\) is the antipodal map. Consider the map \(p:S^2\to \mathbb {R}P^2\) that identifies the antipodal points. Show that all closed 1- and 2-forms on \(\mathbb {R}P^2\) are exact by considering their pullbacks to \(S^{2}\) and applying the above facts.

Find an explicit closed 1-form \(\omega \) on \(\mathbb {R}^2\setminus \{0\}\) which is not exact, and prove both statements.

Find a nonintegrable plane field in \(\mathbb {R}^3\), with a proof.

Let \(f:\mathbb {R}^n\to \mathbb {R}^n\) be a smooth function such that \(f(0)=0\). Show that the graph of \(f\)

and the diagonal

intersect transversely at \((0,0)\) in \(\mathbb {R}^n\times \mathbb {R}^n\) if and only if the derivative \(f_*(0):T_0\mathbb {R}^n\to T_0\mathbb {R}^n\) does not have \(+1\) as an eigenvalue.