Gl(n, R)-invariant tetrad theory and its relation to the purely affine formulation of general relativity

Gl(n, R)-invariant tetrad theory is rewritten so as to show its relation to purely affine formulation of general relativity. As a basic tool for such reformulation the Gl(n, R)-invariant splitting of any connection into a symmetric connection, an antisymmetric second rank tensor and a vector parts was applied. The typical form of the Lagrangian as the square root of a second rank tensor is used. Assuming that the tensor is the symmetric part of the Ricci tensor defined for the symmetric connection part of the basic connection field and that the connection field is flat, equivalence of the obtained Lagrangian and the Lagrangian of the Einstein theory with some constraints on the connection field is shown.