Characterization of the pseudo-symmetries of ideal Wintgen submanifolds of dimension 3

Abstract

Recently, Choi and Lu proved that the Wintgen inequality ? ? H2??? +k, (where ? is the normalized scalar curvature and H2, respectively ??, are the squared mean curvature and the normalized scalar normal curvature) holds on any 3-dimensional submanifold M3 with arbitrary codimension m in any real space form ~M3+m(k) of curvature k. For a given Riemannian manifold M3, this inequality can be interpreted as follows: for all possible isometric immersions of M3 in space forms ~M3+m(k), the value of the intrinsic curvature ? of M puts a lower bound to all possible values of the extrinsic curvature H2 ? ?? + k that M in any case can not avoid to ?undergo" as a submanifold of ?M. From this point of view, M is called a Wintgen ideal submanifold of ~M when this extrinsic curvature H2 ??? +k actually assumes its theoretically smallest possible value, as given by its intrinsic curvature ?, at all points of M. We show that the pseudo-symmetry or, equivalently, the property to be quasi-Einstein of such 3-dimensional Wintgen ideal submanifolds M3 of M~3+m(k) can be characterized in terms of the intrinsic minimal values of the Ricci curvatures and of the Riemannian sectional curvatures of M and of the extrinsic notions of the umbilicity, the minimality and the pseudo-umbilicity of M in ~M.