A Type Of Torus Immersed Into The Unit Sphere Of The Four-dimensional Complex Euclidean Space

In this thesis, we mainly discuss a type of torus immersed into the unit sphere of thefour-dimensional Complex Euclidean Space with the parameter expression: S: r(ξ,η)=(f₁(e^iξ, e^iη), f₂(e^iξ, e^iη), f₃(e^iξ, e^iη), f₄(e^iξ, e^iη)),(a)where (ξ,η)∈[0,2Ï€]×[0,2Ï€], f_k (1≤k≤4) are polynomials of two variables with realcoefficients, and satisfy sum from k=1 to 4|f_k(e^iξ, e^iη)|²=2(b)For the polynomials of two variables, we can express them by f₁(x,y)=sum from k=0 to n sum from l=0 to n a_klx^ky^l,f₂(x,y)=sum from k=0 to n sum from l=0 to n b_kl x^ky^l,(c) f₃(x,y)=sum from k=0 to n sum from l=0 to n c_klx^ky^l,f₄(x,y)=sum from k=0 to n sum from l=0 to n d_klx^ky^l,where a_kl,b_kl, c_kl, d_kl∈R, 1≤k,l≤n are constants.In this paper, considering the feature of the models and applying the Fouriertransformation, the moving frame method and the exterior differential method, we first obtainthe constraint condition equations of the coefficients of f_k (1≤k≤4). Then we concentrateto the case when n=1. Two cases are considered to analyze its existence. And the followingresults are obtained:Theorem2.3.1 Let S be a torus parametrized by (a), where f_k, (1≤k≤4) are allpolynomials of two variables with real coefficients, defined by (c) and satisfy (b). Then Scan't be totally geodesic when n=1.Theorem2.3.2 Let S be a toms parametrized as (a), where,, (1≤k≤4) are allpolynomials of two variables with real coefficients, defined by (c) and satisfy (b). Put (?)=(a₀₀, b₀₀, c₀₀, d₀₀), (?)=(a₀₁, b₀₁, c₀₁, d₀₁),(d) (?)=(a₁₀, b₁₀, c₁₀, d₁₀), (?)=(a₁₁, b₁₁, c₁₁, d₁₁), and (?)=(?)／λ₁, (?)=(?)／λ₂, (?)=(?)／λ₃, (?)=(?)／λ₄, (e)whereλ_i (1≤i≤4) are the length of (?)_i. Let n=1. If it is of constant Kahler angle, then it hasthe following standard form: r=λ₁(?)+λ₂e^iη(?)+λ₃e^iξ(?)+λ₄e^i(ξ+η)(?), (f)Finally we obtain some further geometric properties on S with standard form (f).