**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.*



Can two continuously differentiable functions be made $C^1$ close via a perturbation of the metric?
**Edit.** For an even simpler example, let $f \in C^1(\mathbf{R}^n)$ be non-constant and $h = 0$. Such a metric cannot exist when the constant is so small that $\epsilon < \lvert Df \rvert\_\infty$.


Let $f \in C^1(\mathbf{R}^n)$ be an arbitrary, non-constant function, and $h = -f$. Let moreover the metric $g$ be arbitrary. Then for all euclidean unit vectors $v \in \mathbf{R}^n$, that is vectors with $\sum\_i (v^i)^2 = 1$, one has $D\_v h/g(v,v)^{1/2} - D\_v f = (1/g(v,v)^{1/2} + 1) D\_v h$. Taking absolute values, the supremum of this taken over points $x \in \mathbf{R}^n$ is at least $\lvert D\_v h \rvert\_{\infty} = \lvert D\_v f \rvert\_{\infty}$. In particular if $\epsilon < \sup\_{v \in \mathbf{S}^{n-1}} \lvert D\_v f \rvert\_{\infty}$ then the inequality cannot hold, regardless of the metric.


