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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 Y be an (immersed) oriented hypersurface. At each xY there is a unique (positive) unit normal vector, and hence a well defined Gauss map

lν:YSn-1

assigning to each point xY its unit normal vector, lν(x) . Here Sn-1 denotes the unit sphere, the set of all unit vectors in Rn.

The normal vector, lν(x) is orthogonal to the tangent space to Y at x. We will denote this tangent space by TYx. For our present purposes, we can regard TYx as a subspace of Rn: If tγ(t) is a differentiable curve lying on the hypersurface Y, (this means that γ(t)Y for all t) and if γ(0)=x, then γ(0) belongs to the tangent space TYx. Conversely, given any vector vTYx, we can always find a differentiable curve γ with γ(0)=x, γ(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 Rn. For example suppose that Y=Rn-1 consisting of those points in Rn 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 Sn-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 Sn-1. In other words, it is multiplication by 1/R:

lν(y)=1Ry.

3. Suppose that Y is a right circular cylinder in R3 whose base is the circle of radius r in the x1, x2 plane. Then the Gauss map sends Y onto the equator of the unit sphere, S2, sending a point x into (1/r)π(x) where π : R3R2 is projection onto the x1, x2 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:MRn where M is some open subset of Rn-1 and such that X(M) is some neighborhood of x in Y. Let y1, ..., yn-1 be the standard coordinates on Rn-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:=(Xy1, ... Xyn-1)