Let $M$ be a $n$-dimensional compact Riemannian manifold, and $N$ a smooth submanifold of $M$ of dimension strictly less than $n$.

Denote by $N_{\varepsilon}$ the $\varepsilon$-neighbourhood of $N$ - that is, the set $\{p \in M \ | \ d(x, N) < \varepsilon\}$.

Here $d$ denotes the Riemannian distance.

Let $\text{Vol}$ denote the Riemannian volume measure on $M$.

**Question:**

Define the function $f: [0, \infty) \to \mathbb R_+$ by $f(\varepsilon) := \text{Vol}(N_{\varepsilon})$.

Is it true that

$$\lim_{\varepsilon \to 0+} f’’(\varepsilon)$$

exists? If so, can we find an expression for it in terms of the Riemannian metric on $M$ and the embedding of $N$ in $M$?