a one-sided collar neighbourhood $T$ of $Y$ so that $E\colon Y\times [0, \epsilon) {\rightarrow}T$ is a diffeomorphism. Let $r\colon T {\rightarrow}{\mathbb{R}}$ denote the distance to $Y$. Adapting the proof of the classical Gauss lemma (see [@Lee] or [@Klingenberg] for a precise statement of the classical form) we obtain a decomposition of the metric as $g = dr^2 + g_Y(y,r)$. Further, if we choose any coordinates $\{y^{\beta}\}$ on an open set $U \subset Y$ we may get coordinates on $T$ by extending $y^{\beta}$ to be constant along the integral curves of $\operatorname{grad}r$, and then $(y^{\beta}, r)$ form coordinates on $E(U \times [0, \epsilon)) \subset T$. We will refer to such coordinates as *Fermi coordinates for $Y$*, and in Section \[sec:tracey\] we will see that total convexity implies Fermi coordinates for $Y$ exist on neighbourhoods of the form $U \times [0, \infty)$.

We use Latin indices to index directions in $M$ and consequently these indices range from $0$ to $n$. We use Greek indices to index the directions along $Y$ which range from $1$ to $n$, and a zero or ‘$r$’ to index the direction normal to $Y$.

We now consider the second fundamental form of $Y$ in $M$ and we fix 