| Title:
|
On Deszcz symmetries of Wintgen ideal submanifolds (English) |
| Author:
|
Petrović-Torgašev, Miroslava |
| Author:
|
Verstraelen, Leopold |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
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0044-8753 (print) |
| ISSN:
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1212-5059 (online) |
| Volume:
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44 |
| Issue:
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1 |
| Year:
|
2008 |
| Pages:
|
57-67 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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]. (English) |
| Keyword:
|
submanifolds |
| Keyword:
|
Wintgen inequality |
| Keyword:
|
ideal submanifolds |
| Keyword:
|
Deszcz symmetries |
| MSC:
|
53A10 |
| MSC:
|
53A55 |
| MSC:
|
53B20 |
| MSC:
|
53B25 |
| MSC:
|
53B35 |
| MSC:
|
53C42 |
| idZBL:
|
Zbl 1212.53028 |
| idMR:
|
MR2431231 |
| . |
| Date available:
|
2008-06-06T22:52:46Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/108096 |
| . |
| Reference:
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| Reference:
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