On Levy-Chivita metric connectivity on distribution of planes

In n-dimensional projective space, the distribution of $m$-dimensional planes with a given metric tensor is investigated in the paper. The object of tangent connection is considered, and it is shown that the affine distributed connection can be generalized with a Levi-Civita connection in the case of a holonomic distribution and in the case of a semi-normalized distribution of the 1st kind with the corresponding adaptation of the frame. It is proved that the subobject of the tangent distributed connection can also be covered by the field of the metric tensor, but only in the adapter frame.