6510-2021-08

Identify \(\mathcal {M}(2)\), the set of two-by-two matrices, with \(\mathbb {R}^4\). Let \(SL(2,\mathbb {R})\subset \) \(\mathcal {M}(2)\) be the set of matrices with determinant one. Show that \(SL(2,\mathbb {R})\) is a smooth submanifold and calculate the tangent space \(T_{\mathrm {id}}SL(2,\mathbb {R})\) as a subspace of \(T_{\mathrm {id}}\mathcal {M}(2)=\mathbb {R}^4\).

Let \(N=\{(x,y,z,w)\in \mathbb {R}^4:x^2+y^2=z^2+w^2=1\}\) and \(f(p,q,r):\) \(\mathbb {R}^3\to \mathbb {R}^4\) be given by \(x=p+r\), \(y=q-r\), \(z=p-r\), \(w=p+q\). Show that \(f^{-1}(N)\) an embedded submanifold of \(\mathbb {R}^3\). What is its dimension?

Let \(M\) be a smooth manifold and \(V\) a vector field on \(M\) such that flow for \(V\) is defined for all time. Let \(W\) be another vector field on \(M\) such that \(V\) and \(W\) agree outside of a compact set in \(M\) . Show that \(W\) is defined for all time.

Let \(X\) and \(Y\) be closed sub-manifolds of \(\mathbb {R}^n\). Show that for almost all \(a\in \mathbb {R}^n\) the translate \(X+a=\{x+a\mid x\in X\}\) intersects \(Y\) transversally.

Let \(M\) be a closed, compact manifold and show that if \(\omega \) is a nowhere zero 1-form on \(M\) then the de Rham cohomology group \(H^1_{dR}(M)\) is non-trivial. Use this to show that if there is a submersion from \(M\) to the circle then \(H^1_{dR}(M)\) is non-trivial.

Let \(\phi : G \to H\) be a Lie group homomorphism between Lie groups \(G\) and \(H\). Let \(X\) and \(Y\) be \(\phi \)-related vectors fields on \(G\) and \(H\), respectively, and assume that \(Y\) is left invariant. Is \(X\) left invariant? Either prove this or provide a counterexample.