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Chapter 1
The principal curvatures.
1.1 Volume of a thickened hypersurface
We want to consider the following problem: Let $Y\subset \mathrm{R}^{n}$ be an oriented hyper- surface, so there is a well defined unit normal vector, $l\nu(y)$ , at each point of $Y.$ Let $Y_{h}$ denote the set of all points of the form
$$
y+t_{l\nu}(y)\ ,\ 0\leq t\leq h.
$$
We wish to compute $V_{n}(Y_{h})$ where $V_{n}$ denotes the $n$-dimensional volume. We will do this computation for small $h$, see the discussion after the examples.
Examples in three dimensional space.
1. Suppose that $Y$ is a bounded region in a plane, of area $A$. Clearly
$$
V_{3}(Y_{h})=hA
$$
in this case.
2. Suppose that $Y$ is a right circular cylinder of radius $r$ and height ` with outwardly pointing normal. Then $Y_{h}$ is the region between the right circular cylinders of height ` and radii $r$ and $r+h$ so
$$
V_{3}(Y_{h})\ =\ \pi[(r+h)^{2}-r^{2}]
$$
$$
=\ 2\pi\ell rh+\pi h^{2}
$$
$$
=\ hA+h^{2}\cdot\frac{1}{2r}\cdot A
$$
$$
=\ A(h+\frac{1}{2}\cdot kh^{2})\ ,
$$
where $A=2\pi r$ is the area of the cylinder and where $k=1/r$ is the curvature of the generating circle of the cylinder. For small $h$, this formula is correct, in fact,
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12
{\it CHAPTER 1. THE PRINCIPAL CURVATURES}.
whether we choose the normal vector to point out of the cylinder or into the cylinder. Of course, in the inward pointing case, the curvature has the opposite sign, $k=-1/r.$
For inward pointing normals, the formula breaks down when $h>r$, since we get multiple coverage of points in space by points of the form $y+tl\nu(y)$ .
3. $Y$ is a sphere of radius $R$ with outward normal, so $Y_{h}$ is a spherical shell, and
$$
V_{3}(Y_{h})\ =\ \frac{4}{3}\pi[(R+h)^{3}-R^{3}]
$$
$$
=\ h4\pi R^{2}+h^{2}4\pi R+h^{3}\frac{4}{3}\pi
$$
$$
=\ hA+h^{2}\frac{1}{R}A+h^{3}\frac{1}{3R^{2}}A
$$
$$
=\ \frac{1}{3}\cdot A\cdot[3h+3\frac{1}{R}\cdot h^{2}+\frac{1}{R^{2}}h^{3}],
$$
where $A=4\pi R^{2}$ is the area of the sphere.
Once again, for inward pointing normals we must change the sign of the coefficient of $h^{2}$ and the formula thus obtained is only correct for $h\displaystyle \leq\frac{1}{R}.$
So in general, we wish to make the assumption that $h$ is such that the map
$$
Y\times[\mathrm{O},\ h]\rightarrow \mathrm{R}^{n},\ (y,\ t)\mapsto y+t_{l\nu}(y)
$$
is injective. For $Y$ compact, there always exists an $h_{0}>0$ such that this condition holds for all $h