Riemannian Geometry Part 4

Problem 6.109 Let $(M, g)$ be a Riemannian manifold, $T_{p} M$ the tangent space at $p \in M$ and $T_{p}^{*} M$ its dual space. The musical isomorphisms $b$ and $\sharp$ are defined (see Problem 6.40) by $$ \text { b: } T_{p} M \rightarrow T_{p}^{*} M, \quad X \mapsto X^{\downarrow}, \quad X^{\natural}(Y)=g(X, Y) $$ and its inverse $\omega \mapsto \omega^{\sharp}$, respectively. The gradient of a function $f \in C^{\infty} M$ is defined as $$ \operatorname{grad} f=(\mathrm{d} f)^{\#} \text {. } $$ (i) Prove that $g(\operatorname{grad} f, X)=X f, X \in \mathfrak{X}(M)$. Given local coordinates $\left\{x^{i}\right\}$ : (ii) Compute $\left(\partial / \partial x^{i}\right)^{b}$. (iii) Calculate $\left(\mathrm{d} x^{i}\right)^{\sharp}$. (iv) Write grad $f$ in local coordinates. (v) Verify that in the particular case of $\mathbb{R}^{3}$ equipped with the Euclidean metric, we recover the classical expression of $\operatorname{grad} f$.

Problem 6.110 Let $M$ be a $C^{\infty}$ manifold equipped with a linear connection $\nabla$. Let $\left\{X_{1}, \ldots, X_{n}\right\}$ be a basis of $T_{p} M$, and $\left\{\theta^{1}, \ldots, \theta^{n}\right\}$ its dual basis. The divergence of $Z \in \mathfrak{X}(M)$ is defined by $$ (\operatorname{div} Z)(p)=\sum_{i} \theta^{i}\left(\nabla_{X_{i}} Z\right) $$ (i) Prove that $(\operatorname{div} Z)(p)$ does not depend on the chosen basis. (ii) Show that the divergence of a $C^{\infty}$ field on $\mathbb{R}^{3}$ is the same as the definition given in Advanced Calculus.

Problem 6.111 Let $(M, g)$ be a Riemannian manifold, and let: (a) $\nabla$ be the Levi-Civita connection of $g$. (b) $\operatorname{grad} f$ be the gradient of the function $f \in C^{\infty} M$. (c) $\operatorname{div} X$ be the divergence (see the previous problem) of the vector field $X \in$ $\mathfrak{X}(M)$. For a local field of orthonormal frames $\left(e_{i}\right), i=1, \ldots, n$, we have $\operatorname{div} X=\sum_{i} g\left(\nabla_{e_{i}} X, e_{i}\right)$ (d) $H^{f}$ be the Hessian of $f \in C^{\infty} M$, defined as the second covariant differential $\nabla(\nabla f)$, that is, $$ H^{f}(X, Y)=X Y f-\left(\nabla_{X} Y\right) f, \quad X, Y \in \mathfrak{X}(M) $$ (e) $\Delta f$ be the Laplacian of $f \in C^{\infty} M$, defined by $$ \Delta f=\operatorname{div} \operatorname{grad} f $$ Moreover, suppose $\operatorname{dim} M=3$. Prove the following formulas for $f, h \in C^{\infty} M, X, Y \in \mathfrak{X}(M)$ : 1. $\operatorname{grad}(f h)=f \operatorname{grad} h+h \operatorname{grad} f$. 2. $\operatorname{div}(f X)=X f+f \operatorname{div} X$. 3. $H^{f h}=f H^{h}+h H^{f}+\mathrm{d} f \otimes \mathrm{d} h+\mathrm{d} h \otimes \mathrm{d} f$. 4. $\Delta(f h)=f \Delta h+h \Delta f+2 g(\operatorname{grad} f, \operatorname{grad} h)$.