CHARACTERIZATION OF THE PSEUDO-SYMMETRIES OF IDEAL WINTGEN SUBMANIFOLDS OF DIMENSION 3


Deszcz R., Petrovic-Torgasev M., Şentürk Z., Verstraelen L.

PUBLICATIONS DE L INSTITUT MATHEMATIQUE-BEOGRAD, vol.88, no.102, pp.53-65, 2010 (SCI-Expanded) identifier identifier

Abstract

Recently, Choi and Lu proved that the Wintgen inequality rho <= H-2 - rho(perpendicular to) + kappa, (where rho is the normalized scalar curvature and H-2, respectively rho(perpendicular to), are the squared mean curvature and the normalized scalar normal curvature) holds on any 3-dimensional submanifold M-3 with arbitrary codimension m in any real space form (M) over tilde (3+m)(kappa) of curvature kappa. For a given Riemannian manifold M-3, this inequality can be interpreted as follows: for all possible isometric immersions of M-3 in space forms (M) over tilde (3+m)(kappa), the value of the intrinsic curvature rho of M puts a lower bound to all possible values of the extrinsic curvature H-2 - rho(perpendicular to) + kappa that M in any case can not avoid to "undergo" as a submanifold of (M) over tilde. From this point of view, M is called a Wintgen ideal submanifold of (M) over tilde when this extrinsic curvature H-2 - rho(perpendicular to) + kappa actually assumes its theoretically smallest possible value, as given by its intrinsic curvature rho, 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 M-3 of (M) over tilde (3+m)(kappa) 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) over tilde.