In mathematics, an Eells–Kuiper manifold is a
compactification
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
Compaction may refer t ...
of
by a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
of dimension
, where
, or
. It is named after
James Eells
James Eells (October 25, 1926 – February 14, 2007) was an American mathematician, who specialized in mathematical analysis.
Biography
Eells studied mathematics at Bowdoin College in Maine and earned his undergraduate degree in 1947. Afte ...
and
Nicolaas Kuiper
Nicolaas Hendrik Kuiper (; 28 June 1920 – 12 December 1994) was a Dutch mathematician, known for Kuiper's test and proving Kuiper's theorem. He also contributed to the Nash embedding theorem.
Kuiper studied at University of Leiden in 1937-4 ...
.
If
, the Eells–Kuiper manifold is
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
to the
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has bas ...
. For
it is
simply-connected and has the integral cohomology structure of the
complex projective plane
In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates
:(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...
(
), of the
quaternionic projective plane (
) or of the
Cayley projective plane (
).
Properties
These manifolds are important in both
Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiabl ...
and
foliation theory:
Theorem: ''Let
be a
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
closed manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
(not necessarily
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
) of dimension
. Suppose
admits a
Morse function
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
of class
with exactly three
singular points. Then
is a Eells–Kuiper manifold.''
Theorem:
[.] ''Let
be a compact connected manifold and
a
Morse foliation on
. Suppose the number of
centers
Center or centre may refer to:
Mathematics
* Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of the foliation
is more than the number of
saddle
The saddle is a supportive structure for a rider of an animal, fastened to an animal's back by a girth. The most common type is equestrian. However, specialized saddles have been created for oxen, camels and other animals. It is not kno ...
s
. Then there are two possibilities:''
* ''
, and
is homeomorphic to the sphere
'',
* ''
, and
is an Eells–Kuiper manifold,
or
.''
See also
*
Reeb sphere theorem
In mathematics, Reeb sphere theorem, named after Georges Reeb, states that
: A closed oriented connected manifold ''M'' ''n'' that admits a singular foliation having only centers is homeomorphic to the sphere ''S'n'' and the foliation h ...
References
Foliations
Manifolds
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