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In [[Riemannian geometry]], an '''isoparametric manifold''' is a type of (immersed) [[submanifold]] of [[Euclidean space]] whose [[normal bundle]] is flat and whose [[principal curvatures]] are constant along any [[Parallel transport|parallel]] normal vector field.  The set of isoparametric manifolds is stable under the [[mean curvature flow]].
I'm a 41 years old, married and study at the college (Education Science).<br>In my free time I'm trying to teach myself Japanese. I've been there and look forward to returning anytime soon. I love to read, preferably on my ipad. I like to watch The Vampire Diaries and American Dad as well as docus about nature. I enjoy Sculling or Rowing.<br><br>Here is my webpage ... [http://desktopgrid.hiast.edu.sy/hiastdg/view_profile.php?userid=1434 wordpress backup]
 
== Examples ==
 
A straight line in the plane is an obvious example of isoparametric manifold. Any affine subspace of the Euclidean n-dimensional space is also an example since the principal curvatures of any shape operator are zero.  
Another simplest example of an isoparametric manifold is a sphere in Euclidean space.
 
Another example is as follows.  Suppose that ''G'' is a [[Lie group]] and ''G''/''H'' is a [[symmetric space]] with canonical decomposition
 
:<math>\mathbf{g} = \mathbf{h}\oplus\mathbf{p}</math>
 
of the [[Lie algebra]] '''g''' of ''G'' into a [[direct sum of Lie algebras|direct sum]] (orthogonal with respect to the [[Killing form]]) of the Lie algebra '''h''' or ''H'' with a complementary subspace '''p'''. Then a principal [[group orbit|orbit]] of the [[adjoint representation]] of ''H'' on '''p''' is an isoparametric manifold in '''p'''. Non principal orbits are examples of the so-called '''submanifolds with principal constant curvatures'''. Actually, by Thorbergsson's theorem any complete, full and irreducible isoparametric submanifold of codimension > 2 is an orbit of a s-representation, i.e. an H-orbit as above where the symmetric space ''G''/''H'' has no flat factor.
 
The theory of isoparametric submanifolds is deeply related to the theory of holonomy groups. Actually, any isoparametric submanifold is foliated by the holonomy tubes of a submanifold with constant principal curvatures i.e. a focal submanifold.  The paper '''Submanifolds with constant principal curvatures and normal holonomy groups'''. International J. Math. 2 (1991),167–175 by Heintze, Olmos and Thorbergsson is a very good introduction to such theory. For more detailed explanations about holonomy tubes and focalizations see the book "Submanifolds and Holonomy" by Berndt, Console and Olmos.
 
== References ==
 
* {{cite journal|author=Berndt, J, Console, S, Olmos, C|title=Submanifolds and Holonomy|year=2003|publisher=Chapman and Hall/CRC}}
* {{cite journal|author=Ferus, D, Karcher, H, and Münzner, HF|title=Cliffordalgebren und neue isoparametrische Hyperflächen|journal=Math. Z.|volume=177|year=1981|pages=479–502|doi=10.1007/BF01219082|issue=4}}
* {{cite journal|author=Palais, RS and [[Chuu-Lian Terng|Terng, C-L]]|title=A general theory of canonical forms|journal=Transactions of the American Mathematical Society|volume=300|year=1987|pages=771–789|doi=10.2307/2000369|jstor=2000369|issue=2|publisher=Transactions of the American Mathematical Society, Vol. 300, No. 2}}
* {{cite journal|author=Terng, C-L|title=Isoparametric submanifolds and their Coxeter groups|journal=[[Journal of Differential Geometry]]|year=1985|volume=21|pages=79–107}}
* {{cite journal|author=Thorbergsson, G|title=Isoparametric submanifolds and their buildings|journal=[[Ann. Math.]]|year=1991|volume=133|pages=429–446}}
 
==See also==
 
*[[Isoparametric function]]
 
[[Category:Riemannian geometry]]
[[Category:Manifolds]]
 
 
{{differential-geometry-stub}}

Latest revision as of 23:22, 29 October 2014

I'm a 41 years old, married and study at the college (Education Science).
In my free time I'm trying to teach myself Japanese. I've been there and look forward to returning anytime soon. I love to read, preferably on my ipad. I like to watch The Vampire Diaries and American Dad as well as docus about nature. I enjoy Sculling or Rowing.

Here is my webpage ... wordpress backup

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