D O C U M E N T 5 2 1 U N I F I E D F I E L D T H E O R Y 4 4 9 The symbol [Kronecker delta] is equal to 1 or to 0, depending on whether its in- dices are the same or different. It follows from (3) and (1) that . And from this, it further follows that the metric tensor of the manifold can be expressed in the form: . . . (6) In addition, we set . . . (7) since, according to (6), (7), and (4), the relations are obeyed. We furthermore readily find the equations These equations illustrate how, with the aid of the quantities h, one can go from the usual vector components to the local coordinates and vice versa. An analogous conclusion holds also for tensors of higher rank. We furthermore can consider (h ) and (h ) as second-rank tensors with a local and a usual index. They are connected by the relations as can readily be proven. The tensor theory that is suited to the spatial structure that we are considering here thus has the following properties: Tensors have in general three types of indi- ces. Example: . They are defined through their transformation laws with respect to arbitrary coordinate transformations and to conformal rotations of the lo- cal orthogonal n-tuples. For example, , where the constants form an orthogonal system. From this, the algebraic rules for the tensors follow. Tensors without a Roman index are invariants with respect to coordinate transformations. Furthermore, it fol- lows readily from the relations given that the quantities have tensor character (fundamental tensor). h a h a     . . . (4) = h a h b   ab . . . (5) = A2 h a h A AA = g  g  h a h A = g h a  h a  = g  g    = A a h a gA  h a A  . . . (8) = = A  h a A a . . . (9) = a a  h a g  h a  . . . (10) = h a  gh a . . . (11) = [p. 5a] A s   A s     st x'x' x -----------------A x t   =  st h s 