# Geometric Realizations of Curvature

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.

**Submitted by Anonymous |

**22 / Jul / 2013