In computations for my research I am frequently coming across the contraction $II(\nabla f, \cdot)$ between the second fundamental form $II$ of a surface $M$ and the gradient of a smooth function $f:M \to \mathbb{R}$.
This leads me to believe that there is some nice geometric meaning to this expression, but I'm not sure what it is. Do any of you have a geometrically meaningful interpretation? So far I am just considering it to be like some special kind of gradient, but that's not very satisfying.
Thanks in advance,
Recall that $\langle II(\nabla f, X), N \rangle = \langle S_N \nabla f, X \rangle$, where $X$ is a tangent vector, $N$ the normal vector and $S$ the shape operator. So $II(\nabla f, \cdot)$ essentially tells you how the normal turns along the surface when you walk in the direction where the function $f$ increases the most.
Edit: Also $\langle II(\nabla f, X), N \rangle = \langle S_N X, \nabla f \rangle =df(S_N X)$. So you can also interpret it as a directional derivative of $f$ in the direction of $S_N X$.