6510-2023-08

Let \(\alpha :G\times M\to M\) be an action of a Lie group \(G\) on a \(C^\infty \) manifold \(M\) such that \(\alpha \) is a continuous function. Prove or find a counterexample: if \(\alpha |_{G\times \{x\}}:G\to M\) is a \(C^\infty \) immersion for every \(x\in M\), then the function \(\alpha \) is \(C^\infty \).

For which values of \(n\) is the manifold \(\mathbb {R}P^n\) orientable? Prove that your answer is correct.

Let \(R:S^2\to \mathbb {R}_+\) be a nonconstant \(C^\infty \) function. Show that \(F:S^2\to \mathbb {R}^3\) is defined by \(F(v)=R(v)v\), then there exists a value of \(r\in \mathbb {R}_+\) such that if \(S_r^2\) is the sphere of radius \(r\) centered at 0, then the intersection of \(S_r^2\) with the image of \(F\) is a nonempty finite union of circles.

Let \(V(x,y,z)=\frac {\partial }{\partial x}+y\frac {\partial }{\partial z}\) and \(W(x,y,z)=\frac {\partial }{\partial y}-x\frac {\partial }{\partial z}\) be vector fields on \(\mathbb {R}^3\). Show that there does not exist a surface \(S\subset \mathbb {R}^3\) such that both \(V\) and \(W\) are tangent to \(S\).

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 \(\omega \) be a closed \(n\)-form on a smooth manifold \(M\) . Let \(N\) be a compact, orientable \(n\)-manifold without boundary and \(f:N\to M\) a smooth map with

Show that deRham cohomology group \(H^n_{dR}(M)\) is not trivial.