## Geometric Realizations of Curvature

Author(s):
Miguel Brozos Vázquez, Peter B Gilkey, Stana Nikcevic
Publisher:
Imperial College Press
Year:
2012
ISBN:
ISBN-13 978-1-84816-741-4, ISBN-10 1-84816-741-5
Short description:

This book is focus on the geometric
realizations of curvature. The authors have organized
some of the results in the literature which fall into
this genre. The findings of numerous investigations
in this field are reviewed and presented
in a clear form, including the latest developments
and proofs.

MSC main category:
53 Differential geometry
MSC category:
53B20
Review:

A central area of study in Differential Geometry
is the examination of the relationship between
purely algebraic properties of the Riemannian
curvature tensor and the underlying geometric
properties of the manifold. The decomposition
of the appropriate space of tensors into irreducible
modules under the action of the appropriate structure
group is crucial. This book is focus on the geometric
realizations of curvature. The authors have organized
some of the results in the literature which fall into
this genre. The findings of numerous investigations
in this field are reviewed and presented
in a clear form, including the latest developments
and proofs.

We recall that, given a family of tensors
$\{T_1,\dots ,T_k\}$ on a vector space $V$,
the structure $\left( V,T_1,\dots ,T_k\right)$ is
said to be \emph{geometrically realizable} if there exist
a manifold $M$, a point $P$ of $M$, and an isomorphism
$\phi \colon V\rightarrow T_PM$ such that
$\phi ^{\ast }L_i(P)=T_i$ where $\{L_1,\dots ,L_k\}$
is a corresponding geometric family of tensor fields on $M$.

The book is organized as follows: In Chapter 1 the authors
introduce some notations and state the main results of the book.
They also discuss the basic curvature decomposition results
leading to various geometric realization results in a number
of geometric contexts. The details and proofs can be found
in the rest of the Chapters. Chapter 2 is devoted to
representation theory and in Chapter 3 some results
from differential geometry are presented. In Chapter 4 and 5
the authors work in the real affine and (para)-complex affine
setting respectively. In Chapter 6 and 7
they perform a similar analysis for real Riemannian geometry
and (para)-complex Riemanian geometry. The results in the
(para)-complex and in the complex settings are presented in
parallel. Finally the authors present a list of the main notational
conventions. Following the list a lengthy bibliography is included.
The book concludes with an index.

Reviewer:
Affiliation:
Departamento de Matemática Aplicada, Escuela Técnica Superior de Arquitectura, UPM, Spain