zbMATH — the first resource for mathematics

On Deszcz symmetries of Wintgen ideal submanifolds. (English) Zbl 1212.53028
Summary: It was conjectured in the paper of P. J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken [Arch. Math., Brno 35, No. 2, 115–128 (1999; Zbl 1054.53075)] that, for all submanifolds \(M^n\) of all real space forms \(\tilde M^{n+m}(c)\), the Wintgen inequality \(\rho \leq H^2 - \rho ^\perp + c\) 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 \(\tilde M\). This conjecture was shown there to be true whenever the 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, whose properties are described mainly following S. Haesen and L. Verstraelen [Manuscr. Math. 122, No. 1, 59–72 (2007; Zbl 1109.53020)].

53B25 Local submanifolds
53B35 Local differential geometry of Hermitian and Kählerian structures
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Full Text: EuDML EMIS