Equiaffine characterization of Lagrangian surfaces in ℝ4

Abstract

For non-degenerate surfaces in ℝ4, a distinguished transversal bundle called affine normal plane bundle was proposed in [K. Nomizu and L. Vrancken, A new equiaffine theory for surfaces in ℝ4, Internat. J. Math. 4(1) (1993) 127–165]. Lagrangian surfaces have remarkable properties with respect to this normal bundle, like for example, the normal bundle being Lagrangian. In this paper, we characterize those surfaces which are Lagrangian with respect to some parallel symplectic form in ℝ4.