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In 1960-62, E. K\"ahler developed what looks as a generalization of the exterior calculus, which he based on Clifford rather than exterior algebra. The role of\ the exterior derivative, $du$, was taken by the more comprehensive derivative $\partial u$ ($\equiv dx^{\mu }\vee d_{\mu }u$), where `$\vee $' stands for Clifford product. The $d_{\mu }u$ represents a set of quantities to which he referred as covariant derivative, and for which he gave a long, ad hoc expression. We provide the geometric foundation for this derivative, based on Cartan's treatment of the structure of a Riemannian differentiable manifold without resort to the concept of the so called affine connections. Buried at advanced points in his presentations is the implied statement that $\partial u=du+\ast ^{-1}d$ $u\ast $, the sign at the front of the coderivative term is a matter of whether we include the unit imaginary or not in the definition of Hodge dual, $\ast $. We extract and put together the pieces of theory that go into his derivation of that statement, which seems to have gone unnoticed in spite of its relevance for a quick understanding of what his `K\"ahler calculus'. K\"ahler produced a most transparent

Author

Vargas, Jose G.

Published

2019

Language

EN