The first Bianchi identity in synthetic differential geometry

Abstract

We give a synthetic treatment of the first Bianchi identity both in the style of differential forms and in the style of tensor fields on the lines of Lavendhomme (Basic Concepts of Synthetic Differential Geometry, Kluwer, Dordrecht, 1996). The tensor-field version of the identity is derived from the corresponding one for microcubes, just as we did for the Jacobi identity of vector fields with respect to Lie brackets in our previous paper (J. Theoret. Phys. 36 (1997) 1099–1131). As a by-product we have found out an identity of microcubes corresponding to the classical identity R(X,Y,Z)= ∇ X ∇ YZ− ∇ Y ∇ XZ− ∇ [X,Y]Z of tensor fields, which has largely simplified Lavendhomme's lengthy proof (Basic Concepts of Synthetic Differential Geometry, Kluwer, Dordrecht, 1996, Section 5.3, Proposition 8, pp. 176–180).