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Geometric realizations of curvature
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Description
Rating
Title
Geometric
realizations
of
curvature
Creator
BrozosVazquez, Miguel.
Contributors
Gilkey, Peter B.
Nikcevic, Stana.
World Scientific (Firm)
DescriptionAbstract
A
central
area
of
study
in
Differential
Geometry
is
the
examination
of the
relationship
between
the
purely
algebraic
properties
of the
Riemann
curvature
tensor
and the
underlying
geometric
properties
of the
manifold
. In this
book
, the
findings
of
numerous
investigations
in this
field
of
study
are
reviewed
and
presented
in a
clear
,
coherent
form
,
including
the
latest
developments
and
proofs
.
Even
though
many
authors
have
worked
in this
area
in
recent
years
,
many
fundamental
questions
still
remain
unanswered
.
Many
studies
begin
by
first
working
purely
algebraically
and then
later
progressing
onto the
geometric
setting
and
it
has been
found
that
many
questions
in
differential
geometry
can
be
phrased
as
problems
involving
the
geometric
realization
of
curvature
.
Curvature
decompositions
are
central
to
all
investigations
in this
area
. The
authors
present
numerous
results
including
the
SingerThorpe
decomposition
, the
Bokan
decomposition
, the
Nikcevic
decomposition
, the
TricerriVanhecke
decomposition
, the
GrayHervella
decomposition
and the
De
Smedt
decomposition
. They then
proceed
to
draw
appropriate
geometric
conclusions
from these
decompositions
. The
book
organizes
, in
one
coherent
volume
, the
results
of
research
completed
by
many
different
investigators
over
the
past
30
years
.
Complete
proofs
are
given
of
results
that are
often
only
outlined
in the
original
publications
.
Whereas
the
original
results
are
usually
in the
positive
definite
(Riemannian
setting)
, here the
authors
extend
the
results
to the
pseudoRiemannian
setting
and then
further
, in a
complex
framework
, to
paraHermitian
geometry
as
well
. In
addition
to that,
new
results
are
obtained
as
well
,
making
this an
ideal
text
for
anyone
wishing
to
further
their
knowledge
of the
science
of
curvature
.
DescriptionTable Of Contents
1
.
Introduction
and
statement
of
results
.
1.1
.
Notational
conventions
.
1.2
.
Representation
theory
.
1.3
.
Affine
structures
.
1.4
.
Mixed
structures
.
1.5
.
Affine
Kahler
structures
.
1.6
.
Riemannian
structures
.
1.7
.
Weyl
geometry
I
.
1.8
.
Almost
pseudohermitian
geometry
.
1.9
. The
Gray
identity
.
1.10
.
Kahler
geometry
in the
Riemannian
setting
I
.
1.11
.
Curvature
KahlerWeyl
geometry
.
1.12
. The
covariant
derivative
of the
Kahler
form
I
.
1.13
.
Hyperhermitian
geometry

2
.
Representation
theory
.
2.1
.
Modules
for a
group
G
.
2.2
.
Quadratic
invariants
.
2.3
.
Weyl's
theory
of
invariants
.
2.4
.
Some
orthogonal
modules
.
2.5
.
Some
unitary
modules
.
2.6
.
Compact
Lie
groups

3
.
Connections
,
curvature
, and
differential
geometry
.
3.1
.
Affine
connections
.
3.2
.
Equiaffine
connections
.
3.3
. The
LeviCivita
connection
.
3.4
.
Complex
geometry
.
3.5
. The
Gray
identity
.
3.6
.
Kahler
geometry
in the
Riemannian
setting
II

4
.
Real
affine
geometry
.
4.1
.
Decomposition
of
[symbol]
and
[symbol]
as
orthogonal
modules
.
4.2
. The
modules
[symbol]
,
S[symbol]
and
[symbol]
in
[symbol]
.
4.3
. The
modules
W[symbol]
,
W[symbol]
and
W[symbol]
in
[symbol]
.
4.4
.
Decomposition
of
[symbol]
as a
general
linear
module
.
4.5
.
Geometric
realizability
of
affine
curvature
operators
.
4.6
.
Decomposition
of
[symbol]
as an
orthogonal
module

5
.
Affine
Kahler
geometry
.
5.1
.
Affine
Kahler
curvature
tensor
quadratic
invariants
.
5.2
. The
Ricci
tensor
for a
Kahler
affine
connection
.
5.3
.
Constructing
affine
(para)Kahler
manifolds
.
5.4
.
Affine
Kahler
curvature
operators
.
5.5
.
Affine
paraKahler
curvature
operators
.
5.6
.
Structure
of
[symbol]
as a
GL[symbol]
module

6
.
Riemannian
geometry
.
6.1
. The
Riemann
curvature
tensor
.
6.2
. The
Weyl
conformal
curvature
tensor
.
6.3
. The
CauchyKovalevskaya
theorem
.
6.4
.
Geometric
realizations
of
Riemann
curvature
tensors
.
6.5
.
Weyl
geometry
II

7
.
Complex
Riemannian
geometry
.
7.1
. The
decomposition
of
[symbol]
as
modules
over
[symbol]
.
7.2
. The
submodules
of
[symbol]
arising
from the
Ricci
tensors
.
7.3
.
Parahermitian
and
pseudohermitian
geometry
.
7.4
.
Almost
parahermitian
and
almost
pseudohermitian
geometry
.
7.5
.
Kahler
geometry
in the
Riemannian
setting
III
.
7.6
.
Complex
Weyl
geometry
.
7.7
. The
covariant
derivative
of the
Kahler
form
II
.
Publisher
Imperial College Press
Distributed by World Scientific Pub. Co.
Subject
Geometry.
Geometry, Affine.
Identifier (Full text)
9781848167421
(electronic
bk.)
;
1848167415
;
9781848167414
;
http://www.worldscientific.com/worldscibooks/10.1142/P787#t=toc
Language
eng
Type
Electronic books.
FormatExtent
ix, 252 p. : ill.
Date
c2012
.
RelationIs Part Of
ICP advanced texts in mathematics
v. 6
OCLC number
874497131
CONTENTdm number
221
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