Suppose that we are given a non-negative even function $b\in C^\infty[-1,1]$ satisfying $b(0)=0$, $\sqrt{b(x+y)}\le \sqrt{b(x)}+\sqrt{b(y)}$ for any $x,y\in[-\frac12,\frac12]$. Can we always find a 3-dimensional smooth Riemannian manifold $(M,g)$ and a smooth curve $a:[-\frac12,\frac12]\to M$ parametrized by arc length, such that the square of the distance function $d_g(a(t),a(s))^2=b(t-s)$?

For example, if we set $(M,g)=(\mathbb{R}^3,\|\cdot\|)$, can we find a smooth curve $a:[-\frac12,\frac12]\to \mathbb{R}^3$ parametrized by arc length, such that the square of the distance function $\|a(t)-a(s)\|^2=b(t-s)$?