Some results on traceless decomposition of tensors. (English. Russian original) Zbl 1332.53020

J. Math. Sci., New York 174, No. 5, 627-640 (2011); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 124, No. 1 (2010).
On a Riemannian manifold \(M\) with metric tensor \(g_{ij}\) any tensor \(T\) of type \((0,q)\) has a decomposition \(T=\widetilde{T}+R\), where \(R\) is a sum of \(s<q(q+)/2\) tensors of the form \[ g_{i_{k_\sigma}i_{l_\sigma}}\cdot \overset{(k_\sigma,i_\sigma)}{M}_{i_1,\dots,i_{k_\sigma-1},i_{k_\sigma+1},\dots,i_{l_\sigma-1}, i_{l_\sigma+1},\dots,i_q}, \] where \((i_{k_\sigma},i_{l_\sigma})\) are \(s\) pairs of indices, \(\sigma=1,\dots,s\), such that \(k_\sigma<l_\sigma\) and \(\overset{(k_\sigma,i_\sigma)}{M}\) are tensors of type \((0,q-2)\). Moreover, \(\widetilde{T}\) satisfies the trace condition \(\widetilde{T}_{i_1\cdots i_{k_\sigma}\cdots i_{l_\sigma}\cdots i_q} g^{i_{k_\sigma}i_{l_\sigma}}=0\) for \(\sigma=1,\dots,s\). Similar results hold for mixed tensors \(T\) of type \((p,q)\). The authors here follow investigations of D. Krupka [Linear Multilinear Algebra 54, No. 4, 235–263 (2006; Zbl 1103.15015)] and the author [in: Differential geometry and applications. Proceedings of the 6th international conference, Brno, Czech Republic, August 28–September 1, 1995. Brno: Masaryk University. 45–50 (1996; Zbl 0858.15020)]. For a more general decomposition theorem of this kind occurring in the paper of Krupka, a more constructive proof of the uniqueness of the decomposition is given reducing it to linear equations for the coefficients of the tensors in question.
Another trace condition, called \(F\)-tracial, is defined in terms of an arbitrary \((1,1)\)-tensor \(F\). Furthermore, quaternionic traceless decompositions, generalized decompositions of Ricci and Riemann tensors, of holomorphic projective curvature tensors and Bochner tensors on Kählerian and Hermitian spaces and relations with recurrence are considered. Various examples of such decompositions are included.


53A45 Differential geometric aspects in vector and tensor analysis
53B20 Local Riemannian geometry
53B05 Linear and affine connections
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