6510-2023-01

Define \(F:\mathbb {R}^3\to \mathbb {R}^2\) by \(F(x,y,z)=(x^2+y^2-2,z)\) and let \(S^1=\) \(\{(x,y)\in \mathbb {R}^2\mid x^2+y^2=1\}\) be the unit circle. Show that \(F^{-1}(S^1)\) is a smooth manifold and calculate its dimension.

Let \(V\) be a smooth vector field on \(\mathbb {R}^2\) and assume that outside of a

compact set \(V=\frac {\partial }{\partial x}\) where \(\frac {\partial }{\partial x}\) is the standard horizontal vector field. Show that the flow for \(V\) exists for all time.

Let \(M\) and \(N\) be submanifolds of \(\mathbb {R}^n\) and for all \(a\in \mathbb {R}^n\) let \(M_a=\) \(\{x+a\in \mathbb {R}^n\mid x\in M\}\). Show that \(M_a\) and \(N\) are transverse for almost all choices of \(a\in \mathbb {R}^n\).

Find two 1-forms \(\alpha \) and \(\beta \) on \(\mathbb {R}^2-\{(0,0),(0,2)\}\) that are closed but not exact and have the property that \(\alpha +t\beta \) is not exact for any \(t\in \mathbb {R}\). (Hint: Let \(\iota :S^1\hookrightarrow \mathbb {R}^2\) be inclusion of \(S^1\) into \(\mathbb {R}^2\). Then choose \(\alpha \) and \(\beta \) such that

Let \(M\) be a smooth manifold and \(TM\) its tangent bundle. Show that \(TM\) is orientable.

Let \(G\) be an \(n\)-dimensional Lie group. Show that the tangent bundle \(TG\) is isomorphic as a bundle to the product bundle \(G\times \mathbb {R}^n\).