Surfaces with pointwise 1-type canonical vector field

In the present study, we considered immersed submanifold M⊂R^(n+d) which has pointwise 1-type canonical vector field x^T, i.e. ∆x^T=φ(x^T+C) holds for some non-zero smooth function φ on M and a constant vector C. A pointwise 1-type canonical vector field is called proper if the function φ is non-constant. We obtain some results of rotational surfaces and rotational hypersurfaces in R^(n+d) whose canonical vector fields x^T satisfy this condition. We also give some examples which support the obtained results.