Basis of surface curvature-dependent terms in $6D$

Total derivative terms play an important role in the integration of the conformal anomaly. In four dimensional space $4D$ there is only one such term, namely $\square R$. In the case of six dimensions $6D$ the structure of surface terms is more complicated, and it is useful to construct a basis of linear independent total derivative terms. We briefly review the general scheme of integrating the anomaly and present the reduction of the minimal set of the surface terms in $6D$ from eight to seven. Furthermore, we discuss the comparison with the previously known equivalent reduction based on the general covariance and obtain it also from the conformal symmetry. Our results confirm that the anomaly induced effective action in $6D$ really has a qualitatively new (compared to previously elaborated $2D$ and $4D$ cases) ambiguity, which is parametrized by the two parameters ${\xi }_1$ and ${\xi }_2$.