6510-2024-08

For which \(a > 0\) do the surfaces

have nonempty transverse intersection?

We will identify the set of all real \(n\times n\) matrices with \(\mathbb {R}^{n^2}\) as usual. Show that \[ O(n)=\{M\mid M\text { is an orthogonal }n\times n\text { matrix}\} \] is a submanifold of \(\mathbb {R}^{n^2}\).

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 \(V,W:\mathbb {R}^4\to \mathbb {R}^4\) be vector fields on \(\mathbb {R}^4\) defined by

and

Is there a nonempty surface \(S\subset \mathbb {R}^4\) such that for every \(p\in S\) both \(V(p)\) and \(W(p)\) are tangent to \(S\)? Find such a surface or show it does not exist.

Let \(\omega \) be a compactly supported \(n\)-form on \(\mathbb {R}^n\), \(n\ge 1\). Show that the following statements are equivalent:

1. \[ \int _{\mathbb {R}^n}\omega =0. \]

2. There exists a compactly supported \((n-1)\)-form \(\eta \) on \(\mathbb {R}^n\) such that \(\omega =d\eta \).

   You are allowed to use computations of de Rham cohomology of Euclidean spaces and spheres but not of the compactly supported cohomology of \(\mathbb {R}^n\).

Let \(M\) be a connected smooth manifold. Show that for all \(p,q\in M\) there is a diffeomorphism \(f:M\to M\) such that \(f(p)=q\).