Geodesic
On Deszcz symmetries of Wintgen ideal submanifolds
Archivum mathematicum, Tome 44 (2008) no. 1, pp. 57-67
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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].
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].
Classification : 53A10, 53A55, 53B20, 53B25, 53B35, 53C42
Keywords: submanifolds; Wintgen inequality; ideal submanifolds; Deszcz symmetries 
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Petrović-Torgašev, Miroslava; Verstraelen, Leopold. On Deszcz symmetries of Wintgen ideal submanifolds. Archivum mathematicum, Tome 44 (2008) no. 1, pp. 57-67. http://geodesic.mathdoc.fr/item/ARM_2008_44_1_a6/

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