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

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

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

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:

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