**Definitions:**

We say a smooth Riemannian metric on $\mathbb R^n$ is smoothly equivalent to Lebesgue measure if the Radon Nikodym derivative of the associated Riemannian volume measure with respect to Lebesgue measure is smooth.

Given a smooth Riemannian metric $g$ on $\mathbb R^n$, and a point $x \in \mathbb R^n$, denote by $\gamma_v$ the unit speed geodesic starting at $x$ in the direction of $v$.

For a scalar function $F: \mathbb R^n \to \mathbb R$, denote by $\partial_v {F|}_g$ the scalar function given by $$ \partial_v {F|}_g (x) := \lim_{t \to 0} \frac{F(\gamma_v (t)) - F(x)}{t} $$

Question:Let $f: \mathbb R^n \to \mathbb R$, $h: \mathbb R^n \to \mathbb R$ be $C^1$ functions. For any $\varepsilon > 0$, does there always exist a smooth Riemannian metric $g$ that is smoothly equivalent to Lebesgue measure such that $$ \sup_{v \in S^{n-1}} \left\|\partial_v {h|}_g - \frac{\partial f}{\partial v}\right\|_{C^0} < \varepsilon $$ for each $1 \leq i \leq n$?

*Note: Here $\|\cdot\|_{C^0}$ denotes the sup norm, and $S^{n-1}$ the unit sphere in $\mathbb R^n$ under the Euclidean metric.*