Cut points and critical points of the exponential map

Suppose $ M $ is a complete Riemannian manifold with exponential map $ exp : TM rightarrow M $, and $ S subseteq M $ is an embedded hypersurface with unit normal $ nu $. The normal exponential map is defined by

$$ epsilon: S times mathbb{R} rightarrow M qquad epsilon(x, t) = exp(x, tnu(x)) $$

Let $ delta : M rightarrow mathbb{R} $ be the Riemannian distance function from $ S $ on $ M $:

$$delta(y) = inf{d(y,x): x in S}$$

and define $ U = { (x, t) : (x, t) in S times mathbb{R}, ; delta(epsilon(x,t)) = t } $ and $ D = {(x,t) : Depsilon(x,t) ; textrm{is injective}} $.

It seems reasonable to conjecture that the inclusion $ D subseteq U $ should hold true. Can anyone prove
or disprove this statement?