SEVEN-DIMENSIONAL CONSIDERATIONS OF EINSTEIN’S CONNECTION. II. THE RECURRENCE RELATIONS OF THE SECOND AND THIRD KIND IN 7-g-UFT

We present powerful recurrence relations of the second and third kind which hold in 7-dimensional Einstein's generalized Riemannian manifold X 7. All considerations in this paper are restricted to the first and second classes of X 7 only since the case of the third class, the simplest case, was already studied by many authors. Einstein's connection in a generalized Riemannian manifold X n has been investigated by many authors for lower-dimensional cases n = 2,...,6. In a series of papers, we obtain a surveyable tensorial representation of 7-dimensional Einstein's connection in terms of unified field tensor, with main emphasis on the derivation of powerful and useful recurrence relations which hold in X 7. In [1], which we denote by I in the present paper, we gave a brief survey of Einstein's unified field theory and derive the recurrence relations of the first kind which hold in a general X n. We continue our investigations in this paper and derive a powerful recurrence relations of the second and third kinds which hold in 7-dimensional Einstein's generalized Riemannian manifold X 7. These relations will be used in a subsequent paper to find a tensorial representation of 7-dimensional Einstein's connection. All considerations in this paper are based on the results and symbolism of I. Whenever necessary, they will be quoted in the present paper. In order to obtain a tensorial representation of the 7-dimensional Einstein's connection Γ ν λµ in terms of g λµ , we need powerful recurrence relations of the third kind satisfied by an arbitrary tensor T ωλµ , skew-symmetric in the first two indices. Therefore, we derive these relations in this paper after introducing the recurrence relations of the second kind satisfied by the basic scalars. All considerations in this paper are restricted to n = 7. In 7-g-UFT, there are four cases, that is, the unified field tensor g λµ belongs to (1) the first class, if K 6 ≠ 0 (2) the second class with the first category, if K 2 ≠ 0, K 4 = K 6 = 0