A generalization of the Gaussian mapping of the classical surface theory of Gauss spherical map. Let two: M ,,} R.} + N be an isometric immersion of n-dimensional smooth manifold M "into Euclidean space Also represents the position vector of M ". Using Euclidean parallelism, a unit normal vector Y of M" at point two is translated to the origin of R "ten", its image is located in the unit hypersphere of R .. + " + n- '. In this way, a mapping gY: U M.,} So + p- is induced, where U soil M' 'represents the unit law bundle on M' '. Mapping g, called M " Gaussian spherical mapping. When n = 2 and p = 1, it is the classic Gaussian mapping.
Gauss spherical mapping
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A generalization of the Gaussian mapping of the classical surface theory of Gauss spherical map. Let two: M ,,} R.} + N be an isometric immersion of n-dimensional smooth manifold M "into Euclidean space Also represents the position vector of M ". Using Euclidean parallelism, a unit normal vector Y of M" at point two is translated to the origin of R "ten", its image is located in the unit hypersphere of R .. + "so + n- '. In this way, a mapping gY: U M.,} So + p- is induced, where U soil M' 'represents the unit law bundle on M' '. Mapping g, called M " Gaussian spherical mapping. When n = 2 and p = 1, it is the classic Gaussian mapping.
for Surface in , Define the mapping ,among them Represents the unit normal vector. in case The second basic form of positive definite, then the mapping Is injective, at this time the Gauss map is well defined.
Gaussian mapping is locally differential homeomorphism.
