On willmore surfaces in Sⁿ of flat normal bundle

We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in Sⁿ must be located in some S³ ⊂ Sⁿ, from which we characterize the Clifford torus as the only non-equatorial homogeneous minimal surface in Sⁿ with flat normal bundle, which improves a result of K. Yang. Then we derive that every Willmore two sphere with flat normal bundle in Sⁿ is conformal to a minimal surface with embedded planer ends in R³. We also point out that for a class of Willmore tori, they have a flat normal bundle if and only if they are located in some S³. In the end, we show that a Willmore surface with flat normal bundle must locate in some S⁶.