In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the field of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, an Iwasawa manifold is a compact quotient of a 3-dimensional complex
Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements ' ...
by a
cocompact
Cocompact may refer to:
* Cocompact group action
* Cocompact Coxeter group
* Cocompact embedding
* Cocompact lattice
{{dab ...
,
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
subgroup. An
Iwasawa manifold is a
nilmanifold
In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N/H, the q ...
, of real dimension 6.
Iwasawa manifolds give examples where the first two terms ''E''
1 and ''E''
2 of the
Frölicher spectral sequence In mathematics, the Frölicher spectral sequence (often misspelled as Fröhlicher) is a tool in the theory of complex manifolds, for expressing the potential failure of the results of cohomology theory that are valid in general only for Kähler ma ...
are not isomorphic.
As a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
, such an Iwasawa manifold is an important example of
a compact complex manifold which does not admit any
Kähler metric Kähler may refer to:
;People
*Alexander Kähler (born 1960), German television journalist
*Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and arc ...
.
References
* .
* {{citation , first1=P. , last1 = Griffiths , author1-link=Phillip Griffiths , author2-link=Joe Harris (mathematician), first2=J., last2= Harris , title=Principles of Algebraic Geometry , series=Wiley Classics Library , publisher=Wiley Interscience , year=1994 , isbn=0-471-05059-8 , page=444
Differential geometry
Lie groups
Homogeneous spaces
Complex manifolds