6510-2022-08

Let \(S^2=\{(x,y,z)\in \mathbb {R}^3\mid x^2+y^2+z^2=1\}\) be the 2-sphere and let \(M_a=f_a^{-1}(0)\) where \(a\in \mathbb {R}\) and \(f_a:\mathbb {R}^3\to \mathbb {R}\) is the function \(f_a(x,y,z)=x^2+y^2-z+a\). For what values of \(a\in \mathbb {R}\) are \(S^2\) and \(M_a\) transverse? Justify your answer.

Let \(\mathcal {D}\) be the 2-dimensional distribution in \(\mathbb {R}^3\) spanned by the vector

fields \[ V=\frac {\partial }{\partial x} \quad \text {and}\quad W=x\frac {\partial }{\partial y}+\frac {\partial }{\partial z}. \] Let \(M\) be a smooth 2-manifold and \(\phi :M\to \mathbb {R}^3\) a smooth map such that \(\phi _*(p)(T_pM)\subset \mathcal {D}_{\phi (p)}\) for all \(p\in M\). Show that the derivative \(\phi _*(p):T_pM\to T_{\phi (p)}\mathbb {R}^3\) is not injective for any \(p\in M\).

Let \(X\) and \(Y\) be smooth vector fields on smooth manifolds \(M\) and \(N\) that are \(f\)-related by a smooth map \(f:M\to N\). If \(\psi _t\) is the flow generated by \(X\) and \(\phi _t\) is the flow generated by \(Y\) show that

Let \(G_0\) and \(G_1\) be Lie groups. Then the product \(G_0\times G_1\) is a Lie group and we let \(p_i:G_0\times G_1\to G_i\) be the projection to each coordinate. Let \(X\) be a vector field on \(G_0\times G_1\) and \(Y_0\) and \(Y_1\) vector fields on \(G_0\) and \(G_1\) such that \(X\) and the \(Y_i\) are \(p_i\)-related. Show that \(X\) if left-invariant if and only if both \(Y_{0}\) and \(Y_{1}\) are left-invariant.

Let \(U\subset \mathbb {R}^n\) be open and \(f:U\to \mathbb {R}^n\) a smooth immersion. Then for each \(x\in U\) we have that \(T_xU\) is canonically isomorphic to \(\mathbb {R}^n\) so

we can define a smooth map \(F:U\times \mathbb {R}^n\to \mathbb {R}^n\times \mathbb {R}^n\) by \[ F(x,v)=(f(x),f_*(x)(v)). \] Show that the derivative \(F_*(x,v):T_{(x,v)}(U\times \mathbb {R}^n)\to T_{F(x,v)}(\mathbb {R}^n\times \mathbb {R}^n)\) has positive determinant as a linear map from \(\mathbb {R}^n\times \mathbb {R}^n= \mathbb {R}^{2n}\) to itself. Use this to show that for any smooth manifold \(M\) its tangent bundle \(TM\) is orientable.

1. Construct an explicit \(n\)-form \(\omega \) on \(\mathbb {R}^n\) with compact support such that \(\omega \) is closed but there does not exists an \((n-1)\)-form \(\eta \) with compact support and \(d\eta =\omega \). You can use the existence of bump functions in your construction but you need to prove that your construction works.
2. Let \(M\) be a smooth, compact orientable \(n\)-manifold without bound- ary and let \((U,\phi )\) be a chart for \(M\). Show that there exists a closed \(n\)-form on \(M\) with support in \(U\) that is not exact.