Gauss map

For an oriented regular surface in

R^{3}

, the Gauss map assigns its chosen unit normal

N = \frac{X _{u} \times X _{v}}{∣ X _{u} \times X _{v} ∣} .

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- A conformal Gauss map does not imply a minimal surface Gauss map

A conformal Gauss map does not imply a minimal surface ( $A^{2} = μ I$ )

For a minimal surface, the shape operator has eigenvalues

k, - k

, so

⟨ d N (v), d N (w)⟩ = k^{2} ⟨ v, w ⟩

. Conversely this metric identity only implies that the two principal curvatures have equal squares. A sphere has equal nonzero principal curvatures and satisfies the identity while having nonzero mean curvature. The distinction is between opposite and equal signs, not equality of magnitudes.

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The Gauss map is a mathematical construct used primarily in differential geometry. It associates a surface in three-dimensional space with a unit normal vector at each point of the surface. More specifically, the Gauss map sends each point on a surface to the corresponding point on the unit sphere that represents the normal vector at that point.

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