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The trace decomposition problem. (English) Zbl 0839.15024

The trace decomposition problem is considered, i.e. the problem of existence and uniqueness of a decomposition of a tensor in which one term is traceless and the remaining terms form a linear combination of Kronecker’s \(\delta\)-tensor with traceless coefficients. Two basic theorems are given which provide a complete solution of the trace decomposition problem.
In the particular cases of the tensor spaces of type (1, 2), (1, 3) and (2, 2) explicit decomposition formulas are given. These formulas are applied to the torsion and curvature tensors on a manifold endowed with a linear connection or a metric. It is shown that the trace decompositions lead to the Weyl projective and conformal curvature tensors.

MSC:

15A72 Vector and tensor algebra, theory of invariants
53A55 Differential invariants (local theory), geometric objects
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