# BasicDDI

 BasicDDIis a reserved word in xTras. It is used to label the basic dimensional dependent identities associated to different covariant derivatives.
• Like curvature tensors, the name of the basic DDI associated to a covariant derivative CD is BasicDDICDdim, where dim is the dimension of the manifold of CD.
• The basic DDI is automatically defined when ConstructDDIs is called, but not before.
• The basic DDI is the generalized Kronecker delta with indices, where is the dimension of the manifold of the associated covariant derivative. It is antisymmetric in its first indices, and antisymmetric in its last, and has a pairwise symmetry upon the interchange of these two sets of indices.
• The basic DDI encodes the antisymmetrization of delta functions, which is identically zero in dimensions. This relation is stored in BasicDDIRelations.
• The basic DDI is completely traceless.
• The basic DDI lives in the rectangular Young diagram consisting of two columns and rows, which is zero in dimensions.
Say we have a 3-dimensional manifold with metric g and covariant derivative CD, the basic DDI is given by
This is a rank-8 tensor which is identically zero in 3 dimensions. Its expression in terms of metrics is
The basic DDI is antisymmetric in its first 4 indices, and its last:
It is symmetric under the simultaneous interchange of the first and last set of indices:
It is completely traceless:
Say we have a 3-dimensional manifold with metric g and covariant derivative CD, the basic DDI is given by
 Out[1]=
This is a rank-8 tensor which is identically zero in 3 dimensions. Its expression in terms of metrics is
 Out[2]=
The basic DDI is antisymmetric in its first 4 indices, and its last:
 Out[3]=
 Out[4]=
It is symmetric under the simultaneous interchange of the first and last set of indices:
 Out[5]=
It is completely traceless:
 Out[6]=