Authors: Deszcz, Ryszard ×
Petrovic-Torgasev, Miroslava
Senturk, Zerrin
Verstraelen, Pol #
Issue Date: 2010
Publisher: L'Institut
Series Title: Publications de l'Institut Mathématique vol:88 issue:102 pages:53-65
Abstract: Abstract. Recently, Choi and Lu proved that the Wintgen inequality ρ <
H2−ρ⊥ +k, (where ρ is the normalized scalar curvature and H2, respectively ρ⊥, 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 ρ of M puts a lower bound to all possible values of the extrinsic curvature H2 − ρ⊥ + 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 −ρ⊥ +k actually assumes its theoretically smallest possible value, as given by its intrinsic curvature ρ, 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 M3+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.
ISSN: 0350-1302
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Geometry Section
× corresponding author
# (joint) last author

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