The Gaussian map associates with each point on the surface a unit normal vector at that point. The ends of all such vectors, pending from one point, lie on a sphere of unit radius.
Gaussian map (Gaussian map, spherical map) is a map from a smooth surface in three-dimensional Euclidean space to a unit sphere, at which a surface point is mapped to the unit normal vector at this point. Named after Karl Friedrich Gauss .
Properties
- The Jacobian of the Gaussian map is equal to the Gaussian curvature of the surface at a given point.
Variations and generalizations
- The Gauss map naturally generalizes to the case of a hypersurface in a Euclidean space of arbitrary dimension.
- For a submanifold of a Euclidean space of arbitrary dimension and codimension, a natural analogue of the Gaussian map is a map that associates a point of a manifold with a Grassmannian point corresponding to a tangent space at this point.
Literature
- B.A. Dubrovin, S.P. Novikov, A.T. Fomenko . Modern geometry. - Any edition.
- P.K. Rashevsky . Riemannian geometry and tensor analysis. - Any edition.
- D. Hilbert, S. Cohn-Vossen . Visual geometry. - Any edition.
- Toponogov V.A. Differential geometry of curves and surfaces. - Fizmatkniga, 2012 .-- ISBN 9785891552135 .