Vector-product Surfaces In 3-D Euclidean Spaces

Differential geometry is a subject which has a long history.It attracts us for its blooming influence on other natural science as well.In 3-dimensional Euclidean Spaces,as it is closely to our real life,cause us paying more attention on it.Most researches pay their attention on higher dimensional Euclidean Spaces or un-developable 3-dimensional Euclidean Spaces,using linear operation.In differential geometry on 3-D Euclidean Spaces,many jobs centered on dot calculation like translation surface and product surface.Here,another important operation through geometry learning,which is the vector product.Here,through combining the Vector-product operation and surface together,a new surface is defined,which is called vector-product surface.As vector product operation is regarded as an outer-value,it is not put into consideration for surface classification.According to vector-product operation,this article concentrates its research on classifying the vector-product surface in 3-D Euclidean Spaces,by using invariant.And here come to the conclusion,the vector product surface,with its Gauss curvature equals to zero,if and only if the surface is a plane or just like which is in 3-dimensional Euclidean space.While the vector product surface,with its average curvature equals to zero,if and only if the surface is a plane as well.