Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 36 (1995), No. 2
Demeter Krupka:
The Trace Decomposition Problem
Abstract
The problem of decomposition of mixed tensor spaces by the trace
operation is considered. It is shown that a tensor
$A=\left(A_{k_1k_2\ldots k_q}^{i_1i_2\ldots i_p}\right)$ can always be
expressed as the sum of a traceless term and a linear combination of the
Kronecker's $\delta$-tensor, with traceless coefficients. The uniqueness of
this decomposition is discussed. Particular cases of the trace decomposition
of tensor spaces of type $(1,2)$, $(1,3)$, and $(2,2)$ including the cases
of the torsion and curvature tensors, are considered, and explicit
decomposition formulas are given. This analysis enables us to derive the
Weyl projective, and Weyl conformal curvature tensors.
Key words: Trace, traceless tensor, tensor space decomposition, torsion
tensor, curvature tensor.
MSC 1991: 15A72, 53A55