Minimal Affine Translation Surfaces In The Heisenberg Group Nil₃

The 3-dimensional Heisenberg group Nil3 can be viewed as R3 endowed with the metric and the group operator on Nil3:X1*X2=(x1+x2,y1+y2,z1+z2+1/2(x1x2-x2y1)),Where X1=(x1,y1,z1),X2=(x2,y2,z2)∈R3.Firstly,the expression of Riemannian connection is obtained by using the lie product of the standard basis ｛e1,e2,e3} in Nil3 space.Secondly,based on J.Inoguchi’s research method of minimal translational surface in Nils space,this paper makes a change in the selection of surface baselines:a new set of plane curves φ(x)and φ(x,y)was obtained by affine transformation x=x,y=y+ax(a≠0)to plane curves φ(x)and φ(y).Then affine translation surfaces can be obtained by using the group operator*.Since the group operator in Nil3 space is not commutative,two affine translation surfaces M1(φ,φ)and M2(φ,φ)can be obtained by two plane curves,and the expressions of the first basic quantities,connections and Gaussian curvature and mean curvature corresponding to affine translation surfaces are obtained.In this paper,two classes of minimal affine translation surfaces in Nil3 space are classified,the parametric expressions of each class of minimal affine translation surfaces are determined by solving differential equations,and the corresponding surface graphs are drawn by Mathematica.