Characterization of the Pseudo-symmetries of Ideal Wintgen Submanifolds of Dimension 3

Ryszard Deszcz, Miroslava Petrović-Torgašev, Zerrin Ṣentürk, Leopold Verstraelen

Recently, Choi and Lu proved that the Wintgen inequality $\rho\leq H^2-\rho^\bot+k$, (where $\rho$ is the normalized scalar curvature and $H^2$, respectively $\rho^\bot$, 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 $\widetilde M^{3+m}(k)$ of curvature $k$. 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 $\widetilde M^{3+m}(k)$, the value of the intrinsic curvature $\rho$ of $M$ puts a lower bound to all possible values of the extrinsic curvature $H^2-\rho^\bot+k$ that $M$ in any case can not avoid to ``undergo" as a submanifold of $\tilde M$. From this point of view, $M$ is called a Wintgen ideal submanifold of $\widetilde M$ when this extrinsic curvature $H^2-\rho^\bot+k$ 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 $\widetilde M^{3+m}(k)$ 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 $\widetilde M$.