2 1 4 D O C . 2 1 6 R I E M A N N I A N G E O M E T R Y & P A R A L L E L I S M § 3. Invariants and Covariants. Within the manifold that we are considering, in addition to the tensors and in- variants of Riemannian geometry, which contain the quantities h only in the com- binations given in (3), there are new tensors and invariants, of which we wish to consider only the simplest examples. If one starts with a vector (Aν) at the point (xν), then the two displacements d and d to the neighboring point (xν + dxν) generate the two vectors and . Their difference also has vector character. Then [15] is also a tensor, as is likewise its antisymmetric part, ... .[16] (10) The fundamental meaning of this tensor in the theory developed here can be seen from the following: When this tensor vanishes, the continuum is Euclidean. If, to wit, , then after multiplication by h νb , we obtain . We can therefore set . The field can thus be derived from the n scalar quantities  b . We now choose the coordinates according to the relation . Then all of the Δν αβ vanish, according to (7a), and the h  as well as the g μν become constant.—[17] Since this tensor Λν αβ is furthermore evidently the formally simplest one that is permitted by our theory, it will necessarily be associated with the simplest charac- A  dA  + A  dA  + dA  dA  –       –  A  dx  =       – 1 2 --       –     = 0 2   h   x h a  x h a –       = = 0  x h b  x h b – = [p. 221] h b  x  b =  b x b =