Gaussian figure

In differential geometry , the Gauss map (named after Carl F. Gauß ) maps a surface in Euclidean space onto the unit sphere . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle S^{2}}$

Gauss first wrote on the subject in 1825 and published it in 1827.

definition

The Gaussian map shifts the unit normals of a patch to the fixed origin of the surrounding space.

On a given oriented surface , the Gaussian map is a continuous map , so that a unit vector orthonormal to the surface is at , namely the normal vector at at . ${\displaystyle X\subset \mathbb {R} ^{3}}$ ${\displaystyle N\colon X\to S^{2}}$ ${\displaystyle N(p)}$ ${\displaystyle X}$ ${\displaystyle p\in X}$ ${\displaystyle X}$ ${\displaystyle p}$

properties

The Gaussian map can only be defined globally, i.e. for all , if the surface can be oriented . It can always be defined locally, i.e. on a small piece of the surface. The functional determinant of the Gaussian map is equal to the Gaussian curvature , and the differential of the Gaussian map is called the Weingarten map or form operator . ${\displaystyle p\in X}$

generalization

Analogously to the above definition, the Gauss map for n-dimensional oriented hypersurface in are defined. ${\displaystyle \mathbb {R} ^{n+1}}$

Web links

• Eric W. Weisstein : Gauss Map . In: MathWorld (English).

swell

• Manfredo Perdigão do Carmo: Riemannian Geometry , Birkhäuser, Boston 1992, ISBN 0-8176-3490-8 , p. 129.