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Register a new userGeometric theory of equiaffine curvature tensors
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We present an algebraic investigation of generalized and equiaffine curvature tensors in a given pseudo-Euclidean vector space and study different orthogonal, irreducible decompositions in analogy to the known decomposition of algebraic curvature tensors. We apply the decomposition results to characterize geometric properties of Codazzi structures and relative hypersurfaces; particular emphasis is on projectively flat structures.
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We use curvature decompositions to construct generating sets for the space of algebraic curvature tensors and for the space of tensors with the same symmetries as those of a torsion free, Ricci symmetric connection; the latter naturally appear in relative hypersurface theory.
We examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry. Suppose given an algebraic curvature operator R at a point P of a manifold M and suppose given a real analytic (resp. C-k for finite k at least 2) pseudo-Riemannian metric on M defined near P. We construct a torsion free real analytic (resp. C-k) connection D which is defined near P on the tangent bundle of M whose curvature operator is the given operator R at P and so that D has constant scalar curvature. We show that if R is Ricci symmetric, then D can be chosen to be Ricci symmetric; if R has trace free Ricci tensor, then D can be chosen to have trace free Ricci tensor; if R is Ricci alternating, then D can be chosen to be Ricci alternating.
We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermitian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov-Petrova geometry, Osserman geometry, and curvature homogeneity in terms of geometric realizations.
We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant *-scalar curvature which satisfies the Kaehler condition at the point in question.
We classify algebraic curvature tensors such that the Ricci operator is simple (i.e. the Ricci operator is complex diagonalizable and either the complex spectrum consists of a single real eigenvalue or the complex spectrum consists of a pair of eigenvalues which are complex conjugates of each other) and which are Jacobi--Ricci commuting (i.e. the Ricci operator commutes with the Jacobi operator of any vector).
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