Riemannian Geometry Part 1

Problem 6.39 Prove that on any differentiable manifold $M$ there exists some Riemannian metric. Hint The manifold $M$ is paracompact (see Definitions 1.1).

Problem 6.40 Let $(M, g)$ be a Riemannian $n$-manifold. Prove: (i) Given $\alpha, \beta \in T_{p}^{*} M$ and an orthonormal basis $\left\{e_{i}\right\}, i=1, \ldots, n$, of $T_{p} M$, and denoting by $g^{-1}$ the contravariant metric associated to $g$, one has $$ g^{-1}(\alpha, \beta)=\sum_{i} \alpha\left(e_{i}\right) \beta\left(e_{i}\right) $$ (ii) For $X \in T_{p} M$, one has $$ g^{-1}\left(\alpha, X^{b}\right)=\alpha(X)=g\left(\alpha^{\sharp}, X\right) $$ where $$ \begin{aligned} & \text { b: } T_{p} M \rightarrow T_{p}^{*} M, \quad X^{b}=g(X, \cdot), \quad \sharp: T_{p}^{*} M \rightarrow T_{p} M, \\ & \alpha^{\sharp}=g^{-1}(\alpha, \cdot), \end{aligned} $$ are the musical isomorphisms (named "flat" and "sharp", respectively) associated to $g$.

Problem 6.41 Let $X_{1}$ and $X_{2}$ be the coordinate vector fields for a set of orthogonal coordinates on a surface. Prove that there are isothermal coordinates (also called conformal coordinates) with the same domain of definition and the same coordinate curves (as images) if and only if $$ X_{2} X_{1}\left(\log \frac{g_{11}}{g_{22}}\right)=0 $$ where $g=\sum_{i, j=1}^{2} g_{i j} \mathrm{~d} x^{i} \otimes \mathrm{d} x^{j}$ is the metric.