On the complete decomposition of curvature tensors of Riemannian manifolds with symmetric connection
type: Item
instance of: scholarly article
title: On the complete decomposition of curvature tensors of Riemannian manifolds with symmetric connection
zbMATH Open document ID: 0728.53016
DOI: 10.1007/BF02844767
author: Neda Bokan
publication date: 1990-01-01T00:00:00Z
review text: Let V be a real vector space of dimension n with inner product g and let \({\mathcal R}(V)\) denote the vector space formed by all the algebraic curvature tensors having the same symmetries as the curvature tensor associated to a symmetric connnection on a Riemannian manifold. The main part of the paper consists in deriving complete decompositions of \({\mathcal R}(V)\) under the action of SO(n). The dimensions of the factors are determined and the projections of an element of \({\mathcal R}(V)\) on these factors are derived. The consideration of the norms of these projections leads to (in)equalities for quadratic invariants which may be used in Riemannian geometry to characterize special spaces. Several examples of applications are given with focus on projective curvature tensors and projective transformations and on locally affine symmetric manifolds.
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 53B05
zbMATH DE Number: 4203187
zbMATH Keywords: curvature tensors
zbMATH Keywords: decompositions
zbMATH Keywords: quadratic invariants
zbMATH Keywords: projective curvature tensors
zbMATH Keywords: symmetric manifolds
Wikidata QID: Q115391464
reviewed by: Lieven Vanhecke
MaRDI profile type: Publication
cites work: Almost Hermitian homogeneous structures
cites work: Le spectre d'une variété riemannienne. (The spectrum of a Riemannian manifold)
cites work: Q3721335
cites work: Q3969485
cites work: Invariant theory, old and new
cites work: The sixteen classes of almost Hermitian manifolds and their linear invariants
cites work: On Affine Transformations of a Riemannian Manifold
cites work: Groups of projective transformations and groups of conformal transformations
cites work: Affine transformations in a Riemannian manifold
cites work: Some remarks on tensor invariants of \(O(n), U(n), Sp(n)\)
cites work: A Theorem on the Affine Transformation Group of a Riemannian Manifold
cites work: Partial curvature structures and conformal geometry of submanifolds
cites work: Q4098405
cites work: On the decomposition of generalized K-curvature tensor fields
cites work: Q3324604
cites work: Studies on Riemannian Homogeneous Spaces
cites work: Q5656054
cites work: Q3956040
cites work: Curvature Tensors in Kaehler Manifolds
cites work: Linear Algebra of Curvature Tensors and Their Covariant Derivatives
cites work: Curvature Tensors on Almost Hermitian Manifolds
cites work: Q4747189
cites work: Q4143861
cites work: Q4435456
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 4203187