The Uniqueness of the Zero Tensor under Specific Symmetry Constraints

The analysis reveals that the two seemingly simple symmetry relations, antisymmetry in the first two indices and symmetry in the last two that are actually a highly restrictive combination. The core of the proof is a series of substitutions that lead to the inescapable conclusion that the tensor must be equal to its own negative ( $T_{a b c}=-T_{a b c}$ ). This logical contradiction forces every component of the tensor to be zero, demonstrating that the only mathematical object that can satisfy these constraints is the trivial zero tensor. This problem serves as a powerful illustration of how the properties and form of a tensor can be completely dictated by its symmetry, leading to a unique and often profound result.

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🏗️Structural clarification of Poof and Derivation

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