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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 $\mathit{Y}$ be an (immersed) oriented hypersurface. At each $\mathit{x}\mathrm{\in }\mathit{Y}$ there is a unique (positive) unit normal vector, and hence a well defined Gauss map

$\mathit{l}\mathit{\nu }\mathrm{:}\mathit{Y}\mathrm{\to }{\mathit{S}}^{\mathit{n}\mathrm{-}\mathrm{1}}$

assigning to each point $\mathit{x}\mathrm{\in }\mathit{Y}$ its unit normal vector, $\mathit{l}\mathit{\nu }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ . Here ${\mathit{S}}^{\mathit{n}\mathrm{-}\mathrm{1}}$ denotes the unit sphere, the set of all unit vectors in ${\mathrm{R}}^{\mathit{n}}\mathrm{.}$

The normal vector, $\mathit{l}\mathit{\nu }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ is orthogonal to the tangent space to $\mathit{Y}$ at $\mathit{x}$. We will denote this tangent space by $\mathit{T}{\mathit{Y}}_{\mathit{x}}$. For our present purposes, we can regard $\mathit{T}{\mathit{Y}}_{\mathit{x}}$ as a subspace of ${\mathrm{R}}^{\mathit{n}}$: If $\mathit{t}\mathrm{↦}\mathit{\gamma }\mathrm{\left(}\mathit{t}\mathrm{\right)}$ is a differentiable curve lying on the hypersurface $\mathit{Y}$, (this means that $\mathit{\gamma }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\in }\mathit{Y}$ for all t) and if $\mathit{\gamma }\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathit{x}$, then $\mathit{\gamma }\mathrm{\prime }\mathrm{\left(}\mathrm{0}\mathrm{\right)}$ belongs to the tangent space $\mathit{T}{\mathit{Y}}_{\mathit{x}}$. Conversely, given any vector $\mathit{v}\mathrm{\in }\mathit{T}{\mathit{Y}}_{\mathit{x}}$, we can always find a differentiable curve $\mathit{\gamma }$ with $\mathit{\gamma }\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathit{x}\mathrm{,}$ $\mathit{\gamma }\mathrm{\prime }\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathit{v}$. So a good way to think of a tangent vector to $\mathit{Y}$ at $\mathit{x}$ is as an ""infinitesimal curve“ on $\mathit{Y}$ passing through $\mathit{x}\mathrm{.}$

Examples:

1. Suppose that $\mathit{Y}$ is a portion of an $\mathrm{\left(}\mathit{n}\mathrm{-}\mathrm{1}\mathrm{\right)}$ dimensional linear or affine sub- space space sitting in ${\mathrm{R}}^{\mathit{n}}$. For example suppose that $\mathit{Y}\mathrm{=}{\mathrm{R}}^{\mathit{n}\mathrm{-}\mathrm{1}}$ consisting of those points in ${\mathrm{R}}^{\mathit{n}}$ whose last coordinate vanishes. Then the tangent space to $\mathit{Y}$ 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 $\mathit{Y}$ onto a single point in ${\mathit{S}}^{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{.}$

2. Suppose that $\mathit{Y}$ is the sphere of radius $\mathit{R}$ (say centered at the origin). The Gauss map carries every point of $\mathit{Y}$ into the corresponding (parallel) point of ${\mathit{S}}^{\mathit{n}\mathrm{-}\mathrm{1}}$. In other words, it is multiplication by $\mathrm{1}\mathrm{/}\mathit{R}$:

$\mathit{l}\mathit{\nu }\mathrm{\left(}\mathit{y}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathit{R}}\mathit{y}\mathrm{.}$

3. Suppose that $\mathit{Y}$ is a right circular cylinder in ${\mathrm{R}}^{\mathrm{3}}$ whose base is the circle of radius $\mathit{r}$ in the ${\mathit{x}}^{\mathrm{1}}\mathrm{,}$ ${\mathit{x}}^{\mathrm{2}}$ plane. Then the Gauss map sends $\mathit{Y}$ onto the equator of the unit sphere, ${\mathit{S}}^{\mathrm{2}}$, sending a point $\mathit{x}$ into $\mathrm{\left(}\mathrm{1}\mathrm{/}\mathit{r}\mathrm{\right)}\mathit{\pi }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ where $\mathit{\pi }$ : ${\mathrm{R}}^{\mathrm{3}}\mathrm{\to }{\mathrm{R}}^{\mathrm{2}}$ is projection onto the ${\mathit{x}}^{\mathrm{1}}\mathrm{,}$ ${\mathit{x}}^{\mathrm{2}}$ 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 $\mathit{X}\mathrm{:}\mathit{M}\mathrm{↦}{\mathrm{R}}^{\mathit{n}}$ where $\mathit{M}$ is some open subset of ${\mathrm{R}}^{\mathit{n}\mathrm{-}\mathrm{1}}$ and such that $\mathit{X}\mathrm{\left(}\mathit{M}\mathrm{\right)}$ is some neighborhood of $\mathit{x}$ in $\mathit{Y}$. Let ${\mathit{y}}^{\mathrm{1}}\mathrm{,}$ $\mathrm{\text{...}}\mathrm{,}$ ${\mathit{y}}^{\mathit{n}\mathrm{-}\mathrm{1}}$ be the standard coordinates on ${\mathrm{R}}^{\mathit{n}\mathrm{-}\mathrm{1}}$. Part of the requirement that goes into the definition of parameterization is that the map $\mathit{X}$ be regular, in the sense that its Jacobian matrix

$\mathit{d}\mathit{X}\mathrm{:}\mathrm{=}\mathrm{\left(}\frac{\mathrm{\partial }\mathit{X}}{\mathrm{\partial }{\mathit{y}}^{\mathrm{1}}}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{}\frac{\mathrm{\partial }\mathit{X}}{\mathrm{\partial }{\mathit{y}}^{\mathit{n}\mathrm{-}\mathrm{1}}}\mathrm{\right)}$