Geometric realizations of curvature
Interesting problems involving the geometric realization of curvature occur in several areas of differential geometry. A detailed exposition of recent results and open question on the geometric realization of the curvature in different settings is the subject of this paper. Firstly, the authors treat the geometric realizability of an affine algebraic curvature operator acting on a linear space and of an algebraic curvature tensor on a linear space endowed with a non-degenerate symmetric bilinear form. Then, they consider realization problems related to affine structures in the presence of a pseudo-Riemannian metric. The same problems are treated in the context of almost Hermitian, almost para Hermitian, almost hyper Hermitian geometries and, in particular, Hermitian and para Hermitian structures are considered. In each of the mentioned cases, particular attention to the decomposition of the appropriate space of tensors into irreducible modules under a suitable structure group is given. Furthermore, the authors point their attention to Ivanov-Petrova and Osserman geometries and discuss several questions on curvature homogeneity. The paper, which is carefully written, is equipped with a wide bibliography.
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