On Deszcz symmetries of Wintgen ideal submanifolds

Miroslava Petrović-Torgašev and Leopold Verstraelen

University of Kragujevac, Faculty of Science Department of Mathematics Radoja Domanovića 12, 34000 Kragujevac, Serbia
Catholic University of Leuven Faculty of Science, Department of Mathematics Celestijnenlaan 200B, 3001 Heverlee, Belgium


Abstract: It was conjectured in [26] that, for all submanifolds $M^n$ of all real space forms $\tilde{M}^{n+m}(c)$, the Wintgen inequality $\rho \le H^2 - \rho ^\perp + c$ is valid at all points of $M$, whereby $\rho $ is the normalised scalar curvature of the Riemannian manifold $M$ and $H^2$, respectively $\rho ^\perp $, are the squared mean curvature and the normalised scalar normal curvature of the submanifold $M$ in the ambient space $\tilde{M}$, and this conjecture was shown there to be true whenever codimension $m = 2$. For a given Riemannian manifold $M$, this inequality can be interpreted as follows: for all possible isometric immersions of $M^n$ in space forms $\tilde{M}^{n+m}(c)$, the value of the intrinsic scalar curvature $\rho $ of $M$ puts a lower bound to all possible values of the extrinsic curvature $H^2 - \rho ^\perp + c$ that $M$ in any case can not avoid to “undergo” as a submanifold of $\tilde{M}$. And, from this point of view, then $M$ is called a Wintgen ideal submanifold when it actually is able to achieve a realisation in $\tilde{M}$ such that this extrinsic curvature indeed everywhere assumes its theoretically smallest possible value as given by its normalised scalar curvature. For codimension $m = 2$ and dimension $n > 3$, we will show that the submanifolds $M$ which realise such minimal extrinsic curvatures in $\tilde{M}$ do intrinsically enjoy some curvature symmetries in the sense of Deszcz of their Riemann-Christoffel curvature tensor, of their Ricci curvature tensor and of their conformal curvature tensor of Weyl, which properties will be described mainly following [20].

AMSclassification: Primary: 53B25; Secondary: 53B35, 53A10, 53C42.

Keywords: submanifolds, Wintgen inequality, ideal submanifolds, Deszcz symmetries