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{\it 1.2. THE GAUSS MAP AND THE WEINGARTEN MAP}.
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1.2 The Gauss map and the Weingarten map.
In order to state the general formula, we make the following definitions: Let $Y$ be an (immersed) oriented hypersurface. At each $x\in Y$ there is a unique (positive) unit normal vector, and hence a well defined Gauss map
$$
l\nu:Y\rightarrow S^{n-1}
$$
assigning to each point $x\in Y$ its unit normal vector, $l\nu(x)$ . Here $S^{n-1}$ denotes the unit sphere, the set of all unit vectors in $\mathrm{R}^{n}.$
The normal vector, $l\nu(x)$ is orthogonal to the tangent space to $Y$ at $x$. We will denote this tangent space by $TY_{x}$. For our present purposes, we can regard $TY_{x}$ as a subspace of $\mathrm{R}^{n}$: If $t\mapsto\gamma(t)$ is a differentiable curve lying on the hypersurface $Y$, (this means that $\gamma(t)\in Y$ for all {\it t}) and if $\gamma(0)=x$, then $\gamma'(0)$ belongs to the tangent space $TY_{x}$. Conversely, given any vector $v\in TY_{x}$, we can always find a differentiable curve $\gamma$ with $\gamma(0)=x, \gamma'(0)=v$. So a good way to think of a tangent vector to $Y$ at $x$ is as an ``infinitesimal curve'' on $Y$ passing through $x.$
Examples:
1. Suppose that $Y$ is a portion of an $(n-1)$ dimensional linear or affine sub- space space sitting in $\mathrm{R}^{n}$. For example suppose that $Y=\mathrm{R}^{n-1}$ consisting of those points in $\mathrm{R}^{n}$ whose last coordinate vanishes. Then the tangent space to $Y$ at every point isjust this same subspace, and hence the normal vector is a constant. The Gauss map is thus a constant, mapping all of $Y$ onto a single point in $S^{n-1}.$
2. Suppose that $Y$ is the sphere of radius $R$ (say centered at the origin). The Gauss map carries every point of $Y$ into the corresponding (parallel) point of $S^{n-1}$. In other words, it is multiplication by $1/R$:
$$
l\nu(y)=\frac{1}{R}y.
$$
3. Suppose that $Y$ is a right circular cylinder in $\mathrm{R}^{3}$ whose base is the circle of radius $r$ in the $x^{1}, x^{2}$ plane. Then the Gauss map sends $Y$ onto the equator of the unit sphere, $S^{2}$, sending a point $x$ into $(1/r)\pi(x)$ where $\pi$ : $\mathrm{R}^{3}\rightarrow \mathrm{R}^{2}$ is projection onto the $x^{1}, x^{2}$ plane.
Another good way to think of the tangent space is in terms of a local
parameterization which means that we are given a map $X:M\mapsto \mathrm{R}^{n}$ where $M$ is some open subset of $\mathrm{R}^{n-1}$ and such that $X(M)$ is some neighborhood of $x$ in $Y$. Let $y^{1}, \ldots, y^{n-1}$ be the standard coordinates on $\mathrm{R}^{n-1}$. Part of the requirement that goes into the definition of parameterization is that the map $X$ be regular, in the sense that its Jacobian matrix
$$
dX:=(\frac{\partial X}{\partial y^{1}},\ \cdots\ \frac{\partial X}{\partial y^{n-1}})
$$
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