Chapter 15

On the \(n\)-sphere \(S^{n}\) find coordinates corresponding to (i) stereographic projection, (ii) spherical polars.

Show that the real projective \(n\)-space \(P^{\text {* }}\) defined in Example \(10.15\) as the set of straight lines through the origin in \(\mathbb{R}^{n+1}\) is a differentiable manifold of dimension \(n\), by finding an atlas of compatible charts that cover it.

Let \(\mathbb{R}^{\prime}\) be the marifold consisting of \(\mathbb{R}\) with differentiable structure generated by the chart \(\left(R ; y=x^{3}\right)\). Show that the identity map id \(\mathbb{R}: \mathbb{R}^{\prime} \rightarrow \mathbb{R}\) is a differentiable homeomorphism, which is not a diffeomorphism.

Show that the set of real \(m \times m\) matrices \(M(m, n ; \mathbb{R})\) is a manifold of dimension \(m n\). Show that the matrix multiplication map \(M(m, k ; \mathbb{R}) \times M(k, n ; \mathbb{R}) \rightarrow M f(m, n ; \mathbb{R})\) is differentiable.

Show that the curve $$ 2 x^{2}+2 y^{2}+2 x y=1 $$ can be converted by a rotation of axcs to the standand form for an ellipse $$ x^{\prime 2}+3 y^{2}=1 $$ If \(x^{\prime}=\cos \psi, v^{\prime}=\frac{1}{\sqrt{3}} \sin \psi\) is used as a parametrization of this curve, show that $$ x=\frac{1}{\sqrt{2}}\left(\cos \psi+\frac{1}{\sqrt{3}} \sin \psi\right), \quad y=\frac{1}{\sqrt{2}}\left(-\cos \psi+\frac{1}{\sqrt{3}} \sin \psi\right) $$ Compute the components of the tangent vector $$ X=\frac{\mathrm{d} x}{\mathrm{~d} \psi} \partial_{x}+\frac{\mathrm{d} y}{\mathrm{~d} \psi} \partial_{y} $$ Show that \(X(f)=(2 / \sqrt{3})\left(x^{2}-y^{2}\right)\).

Show that the tangent space \(T_{(\rho q)}(M \times N)\) at any pount \((p, q)\) of a product manifold \(M \times N\) is naturally isomorphic to the dircct sum of tangent spaces \(T_{p}(M) \oplus T_{q}(N)\)

Express the vector field \(\partial_{\varphi}\) in polar coordinates \((\theta, \phi)\) on the unit 2 -sphere in terms of stereographic coordinates \(X\) and \(Y\).

Is the map \(\alpha: \mathbb{R} \rightarrow \mathbb{R}^{2}\) given by \(x=\sin t, y=\sin 2 t\) (1) an immersion, (ii) an cmbedded submanifold?

Show that the map \(\alpha: \hat{R}^{2} \rightarrow \mathbb{R}^{3}\) defined by $$ u=x^{2}+y^{2}, \quad v=2 x y, \quad w=x^{2}-y^{2} $$ is an immersion Is it an embedded submanifold?

Show that the Jacobi identity can be written $$ \mathcal{L}_{[x, r]} Z=\mathcal{L}_{x} \mathcal{L}_{r} Z-\mathcal{L}_{r} \mathcal{L}_{x} Z $$ and this property extends to all tensors \(T\) :

Let \(\alpha: M \rightarrow N\) be a diffeomerphism between manifolds \(M\) and \(N\) and \(X\) a vector field on \(M\) that generates a local one-parameter group of transformations \(\sigma_{t}\) on \(M\). Show that the vector field \(X^{\prime}=\alpha_{*} X\) on \(N\) generates the local flow \(\sigma_{i}^{\prime}=\alpha \circ \sigma_{t} \circ \alpha^{-1}\).

For any real positive number \(n\) show that the vector field \(X=x^{n} \partial_{x}\) is differentiable on the manifold \(\mathbb{R}^{+}\)consisting of the positive real line \([x \in \mathbb{R} \mid x>0]\). Why is this not true in general on the enture real line \(\mathrm{R}\) ? As done for the case \(n=2\) in Example 15,13, find the maximal one-parameter subgroup \(\sigma_{i}\) generated by this vector field at any point \(x>0\).

On the manifold \(\mathbb{R}^{2}\) with coordinates \((r, y)\), let \(X\) be the vector field \(X=-y \partial_{x}+\) \(x \partial_{y}\). Determine the integral curve through any point \((x, y)\), and the one-parameter group generated by \(X\). Find coordinates \(\left(x^{\prime}, y^{\prime}\right)\) such that \(X=\partial_{x}\),

Show that the Lie derivative \(\mathcal{L}_{x}\) commules with all operations of contraction \(C_{t}^{y}\) on a tensor field \(T\), $$ \mathcal{L}_{X} C_{J} T=C_{j} \mathcal{L}_{\mathrm{Y}} T $$