A characteristic feature of the surfaces with constant Gaussian torsion in E4

It is known that every two-dimensional Riemannian manifold 𝑀2 with Gaussian curvature of constant signs has recurrent Riemannian — Chrictoffel curvature tensor 𝑅. The following equality holds: ∇𝑅 = d ln |𝐾| ⊗ ⊗ 𝑅, where is Riemannian metric 𝑀2, ∇ is Riemannian connection. Let 𝐸4 be 4-dimensional Euclidean space with Cartesian coordinates (𝑥1, 𝑥2, 𝑥3, 𝑥4), 𝐹2 is two-dimensional surface in 𝐸4 given by vector equation ⃗𝑟(𝑢1, 𝑢2) = {𝑥1(𝑢1, 𝑢2), 𝑥2(𝑢1, 𝑢2), 𝑥3(𝑢1, 𝑢2), 𝑥4(𝑢1, 𝑢2)}, (𝑢1, 𝑢2) ∈ 𝑈, 𝑥𝑎(𝑢1, 𝑢2) ∈ 𝐶∞(𝑈), = 1,..., 4. The properties of surfaces 𝐹2 with nonzero Gaussian torsion { ̸= 0 in Euclidean space 𝐸4 are studied in this article. Let 𝑅⊥ be normal curvature tensor of 𝐹2 ⊂ 𝐸4, is normal connection, ∇ = ∇ ⊕ is connection of van der Waerden — Bortolotti. Normal curvature tensor 𝑅⊥ ̸= 0 is called parallel if ∇𝑅⊥ ≡ 0. Normal curvature tensor 𝑅⊥ ̸= 0 is called recurrent (in connection ∇) if there exists 1-form on 𝐹2 such that ∇𝑅⊥ = ⊗ 𝑅⊥. The following statement is proved in this article. A surface 𝐹2 with nonzero Gaussian torsion { ̸= 0 in 𝐸4 has recurrent normal curvature tensor 𝑅⊥: ∇𝑅⊥ = d ln |{| ⊗ 𝑅⊥. The characteristic feature of 2-dimensional surfaces 𝐹2 with constant Gaussian torsion { ≡ const ̸= 0 in 4-dimensional Euclidean space 𝐸4 was obtained in this article. It was proved that surface 𝐹2 ⊂ 𝐸4 has constant Gaussian torsion { ≡ ≡ const ̸= 0 if and only if normal curvature tensor 𝑅⊥ ̸= 0 is parallel in connection of van der Waerden — Bortolotti.