Chapter 15

Let $\alpha :M\to N$ 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 $X^{\prime }={\alpha }_{\ast }X$ 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 $(x^{\prime },y^{\prime })$ 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 $T$, $${\mathcal{L}}_X{C}_{J}T=C_j{\mathcal{L}}_{\mathrm{Y}}T$$