# Second derivative of the volume of the $\varepsilon$-neighbourhood of a submanifold

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 $$M$$?

• You may want to check Hermann Weyl's "On the Volume of Tubes", jstor.org/stable/2371513?seq=1#metadata_info_tab_contents. If $M=\mathbb R^n$, then the volume of $N_\varepsilon$ is a degree $n$ polynomial in the variable $\varepsilon$ whose coefficients are mixed volumes of $N$ (up to constant multiples). Weyl gives an explicit formula for the mixed volumes. Aug 3, 2021 at 4:15
• Wow that’s more complicated than expected. I would hope that taking the limit makes things somewhat easier so we might be able to avoid explicit expressions (for fixed $\varepsilon > 0$, that is). Aug 3, 2021 at 4:23
• I should have mentioned that the point of Weyl's paper is that the mixed volumes are independent of the embedding, i.e. they are invariants of the metric on $N$, which is quite remarkable. See also en.wikipedia.org/wiki/Weyl%27s_tube_formula. Aug 3, 2021 at 4:29

I guess we are interested in closed submanifolds; otherwise the question does not have much sense. Suppose $$k=\mathrm{codim}\, N$$.
Note that $$f(\varepsilon)=O(\varepsilon^k)$$, so $$f''(0)=0$$ if $$k\ge 3$$. It remains to consdier two cases $$k=2$$ and $$k=1$$.
Given a normal vector $$v$$ to $$N$$ denote by $$\rho(v)$$ the jacobian of the expontential map from the normal bundle $$E$$ to $$N$$ to $$M$$. Note that $$f(\varepsilon)=\int\limits_{x\in N}\ \int\limits_{v\in B_\varepsilon\subset E_x} \rho(v).$$ Note that $$\rho$$ is smooth and $$\rho(0)=1$$.
If $$k=2$$, then it follows that $$f''(0)=2\cdot\pi\cdot \mathrm{vol}\,N$$.
For $$k=1$$, we get $$f''(0)=0$$,