**Context.** I'm looking for a "natural" definition of subdifferentials on riemannian manifolds.

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Given a function $F:\mathbb R^m \to \mathbb R$, its Fréchet-subdifferential at a point $w \in \mathbb R^m$, denoted $\partial F(w)$, is defined by to be the following (possible empty) subset of $\mathbb R^m$
$$
\partial F(w) \mathrel{:=} \left\{w^\star \in \mathbb R^m \mid \liminf_{w' \to w}\frac{f(w') - f(w) - \langle w^\star,w'-w\rangle}{\|w'-w\|} \ge 0\right\}.
$$

Note that if $F$ is convex, then this definition reduces to the familiar definition $\partial F(w) = \{w^\star \mid f(w') - f(w) \ge \langle w^\star,w'-w\rangle \;\forall w' \in \mathbb R^m\}$.

Extension to riemannian manifolds
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Now, let $M$ be a riemannian submanifold of $\mathbb R^m$ and let $f:M \to \mathbb R$ be the restriction of $F$ on $M$.

>**Question.** What is a natural way to define $\partial f$, so that it is "compatible" with the differential structure on $M$?

My attempt
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>$\partial f(w) \mathrel{:=} \{P_{T_wM}(w^\star) \mid w^\star \in \partial F(w)\}$,

where $P_{T_wM}:\mathbb R^m \to T_wM$ is the projection operator from $\mathbb R^m$ to the tangent space $T_wM$ of $M$ at $w$. This attempt is motivated by the fact that the ordinary gradient of $f$ (in case $f$ is differentiable), namely $\nabla f(w) \mathrel{:=} P_{T_wM}(\nabla F(w))$.

However

- Is my proposal natural enough ?
- Has it already been proposed / studied elsewhere ?

Any help / references would be greatly appreciated.