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In the last few years we have seen an increase interest on gravitationalwaves due to recent and striking experimental results confirming Einstein'sgeneral relativity once more. From the field theory point of view, gravitydescribes the propagation of self-interacting massless spin-2 particles. Theycan be identified with metric perturbations about a given background metric.Since the metric is a symmetric tensor, the massless spin-2 particles presentin the Einstein-Hilbert (massless Fierz-Pauli) theory are naturally describedby a symmetric rank-2 tensor. However, this is not the only possible consistentmassless spin-2 theory at linearized level. In particular, if we add a massterm, a new one parameter $(a_1)$ family of models ${\cal L}(a_1)$ shows up.They consistently describe massive spin-2 particles about Einstein spaces interms of a non-symmetric rank-2 tensor. Here we investigate the masslessversion of ${\cal L}(a_1)$ in a curved background. In the case $a_1=-1/12$ weshow that the massless spin-2 particles consistently propagate, at linearizedlevel, in maximally symmetric spaces. A similar result is obtained otherwise$(a_1 \ne -1/12)$ where we have a non-symmetric scalar-tensor massless model.The case of partially massless non-symmetric models is also investigated.

Note on massless and partially massless spin-2 particles in a curved background via a nonsymmetric tensor