Variational Principles in Teleparallel Gravity Theories

We study the variational principle and derivation of field equations for different classes teleparallel gravity theories, using both their metric-affine covariant tetrad formulations. These theories have in common that addition to or metric, they employ a flat connection as additional variable, but differ by presence absence torsion nonmetricity this independent connection. Besides underlying geometric formulation metric fundamental one has choices introduce conditions vanishing curvature, nonmetricity, either imposing them priori correspondingly restricting variation action when are derived, Lagrange multipliers. Special care must be taken, since these form non-holonomic constraints. Here we show explicitly all aforementioned approaches equivalent, same set is obtained, independently choice procedure. further discuss consequences arising from diffeomorphism invariance gravitational action, how establish relations between equations.