Let $(M,g)$ be a connected complete smooth Riemannian manifold with $\operatorname{Ric}_g \geq 0$. Let $\gamma: \mathbb{R} \to M$ be a unit-speed line, and define the forward Busemann function
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\begin{align*}
b_+: M &\to \mathbb{R},\\
x &\mapsto \lim_{t \to \infty}\bigl(t - d_g(x,\gamma(t))\bigr).
\end{align*}
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If $b := b_+$, then $b \in C^\infty(M)$, $|\nabla b|_g = 1$ on $M$, and
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\begin{align*}
\operatorname{Hess} b = 0
\end{align*}