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In this note we highlight a common origin for many ubiquitous geometricstructures, as well as several new ones by using only the functors ofdifferential calculus in A.M Vinogradov's original sense, adapted to specialclasses of (graded) commutative algebras. Special attention is given to theparticularly simple cases of diole and triole algebras and we show the latterenvironment is the appropriate one to describe calculus in vector bundles inthe presence of a vector-valued fiber metric.

Differential Calculus over Graded Commutative Algebras and Vector Bundles with Inner Structures