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In 21 and 22, we proved the Kastler-Kalau-Walze type theorem for the J-twist DJ of the Dirac operator on 3-dimensional, 4-dimensional and 6-dimensional almost product Riemannian spin manifold with boundary. In this paper, we generalize our previous conclusions and establish the proof of the general Kastler-Kalau-Walze type theorem for the J-twist DJ of the Dirac operator on even-dimensional almost product Riemannian spin manifold with boundary.
Liu et al. (Mon,) studied this question.