Geometric partial differentiability on manifolds: the tangential derivative and the chain rule

TL;DR

This paper introduces the tangential derivative, a diffeomorphism-invariant directional derivative on manifolds, and characterizes when the chain rule applies, extending classical derivatives to a broader geometric context.

Contribution

It defines the tangential derivative invariant under diffeomorphisms and characterizes chain rule applicability on manifolds, broadening the scope of directional derivatives.

Findings

Tangential derivative is invariant under diffeomorphisms.

Equivalent to classical directional derivative for Lipschitz functions.

Characterization of pairs of functions satisfying the chain rule.

Abstract

We define the tangential derivative, a notion of directional derivative which is invariant under diffeomorphisms. In particular this derivative is invariant under changes of chart and is thus well-defined for functions defined on a differentiable manifold. This notion is weaker than the classical directional derivative in general and equivalent to the latter for Lipschitz continuous functions. We characterize also the pairs of tangentially differentiable functions for which the chain rule holds.