Volume of neighborhood of the curve

Question

Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|<r\}$. How can i show that the volume of $B_r$ is equal to $br^{(n-1)}$ for $r <\varepsilon$ for some $\varepsilon>0$. And how can I find $b$? It is intuitively obvious. But how can one prove it? I have no idea.

Answer 1

It seems that the answer is $|B_r| = LV_r$, where $L$ is the length of the curve $\gamma$ and $V_r$ is the volume of the ball with radius $r$ in $\mathbb R^{n-1}$. This is a bit odd to me because $B_r$ is not isometric to the cylinder, so before the calculation I thought I would get something related to the curvature of $\gamma$.

The point is to find a good parametrization of $B_r$. Let $\{\dot \gamma (t), v_1(t), \cdots v_{n-1}(t)\}$ be orthonormal frame. That is

$$\langle \dot\gamma, v_i\rangle = 0, \ \ \langle v_i , v_j\rangle = \delta_{ij} \ \ \ \text{ for all } t\ .$$

Moreover we choose $v_i$ in such a way that $\frac{\partial v_i}{\partial t} = -\langle v_i , \ddot\gamma \rangle \dot\gamma$. (Abstractly, I consider the Gauss map $G(t)= \dot\gamma(t) \in \mathbb S^{n-1}$ and forms $v_i$ by parallel transport along the curve $G(t)$ on $\mathbb S^{n-1}$). Then define

$$\phi :[0,L] \times B(r) \to B_r,\ \ \phi(t, \vec x) = \gamma(t) + x^i v_i(t)\ .$$

where $B(r)$ is the ball with radius $r$ in $\mathbb R^{n-1}$. Then

$$\partial_t \phi = \dot\gamma + x^i \partial_t v_i = \dot\gamma - x^i h_i \dot\gamma = (1-\langle \vec x , \vec h\rangle)\dot\gamma,\ \ \ \partial_i\phi = v_i$$

where $\vec h := (h_1, \cdots h_{n-1})$ and $h_i := \langle v_i, \ddot\gamma \rangle$. This implies $J\phi = (1- \langle \vec x, \vec h\rangle)$ and

$$|B_r| = \int_0^L \int_{B(r)} (1- \langle \vec x, \vec h\rangle) d\vec x dt = LV_r - \int_0^L \int_{B(r)} \langle \vec x, \vec h\rangle d\vec x = LV_r$$

(The integral on $\langle \vec x, \vec h\rangle$ is zero as it is an "odd function").

Similar question can be asked for a submanifolds $\Sigma$ in $\mathbb R^n$. In this case the curvature of $\Sigma$ plays role and for example, for surfaces in $\mathbb R^3$, we have (given in http://golem.ph.utexas.edu/category/2010/03/intrinsic_volumes_for_riemanni.html)

$$|B_r(\Sigma)| = 2|\Sigma|r + \frac{4\pi}{3} \chi(\Sigma) r^3\ .$$