On Deszcz symmetries of Wintgen ideal submanifolds

Miroslava Petrović-Torgašev; Leopold C. A. Verstraelen

Archivum Mathematicum (2008)

  • Volume: 044, Issue: 1, page 57-67
  • ISSN: 0044-8753

Abstract

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It was conjectured in [26] that, for all submanifolds M n of all real space forms M ˜ n + m ( c ) , the Wintgen inequality ρ ≤ H 2 - ρ ⊥ + c is valid at all points of M , whereby ρ is the normalised scalar curvature of the Riemannian manifold M and H 2 , respectively ρ ⊥ , are the squared mean curvature and the normalised scalar normal curvature of the submanifold M in the ambient space 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 M ˜ n + m ( c ) , the value of the intrinsic scalar curvature ρ of M puts a lower bound to all possible values of the extrinsic curvature H 2 - ρ ⊥ + c that M in any case can not avoid to “undergo” as a submanifold of 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 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 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].

How to cite

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Petrović-Torgašev, Miroslava, and Verstraelen, Leopold C. A.. "On Deszcz symmetries of Wintgen ideal submanifolds." Archivum Mathematicum 044.1 (2008): 57-67. <http://eudml.org/doc/250441>.

@article{Petrović2008,
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].},
author = {Petrović-Torgašev, Miroslava, Verstraelen, Leopold C. A.},
journal = {Archivum Mathematicum},
keywords = {submanifolds; Wintgen inequality; ideal submanifolds; Deszcz symmetries; Wintgen inequality; ideal submanifold; Deszcz symmetry},
language = {eng},
number = {1},
pages = {57-67},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On Deszcz symmetries of Wintgen ideal submanifolds},
url = {http://eudml.org/doc/250441},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Petrović-Torgašev, Miroslava
AU - Verstraelen, Leopold C. A.
TI - On Deszcz symmetries of Wintgen ideal submanifolds
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 1
SP - 57
EP - 67
AB - 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].
LA - eng
KW - submanifolds; Wintgen inequality; ideal submanifolds; Deszcz symmetries; Wintgen inequality; ideal submanifold; Deszcz symmetry
UR - http://eudml.org/doc/250441
ER -

References

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