Let $F:\mathbb R^n \to \mathbb R$ be a "sufficiently regular" function. For any $k \ge 1$ and $x \in \mathbb R^n$, define $$ \alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(x)\|^k}\right). $$ It is well-known that $$ \alpha_1(x)= \dfrac{\nabla F(x)^\top \nabla^2 F(x) \nabla F(x)-\|\nabla F\|^2 \mathop{\rm trace}(\nabla^2 F(x))}{\|\nabla F(x)\|^2} $$ is the mean curvature to the surface $S := \{x \in \mathbb R^n \mid F(x) = 0\}$, at the point $x$.
Question. Is there a geometric intepration of $\alpha_2(x)$ ?
Motivation: density of random variables via Malliavin calculus
Let $\{W(h) \mid h \in H\}$ be an isonormal Gaussian process and let $F=f(W(h_1),\ldots,W(h_n))$ be a "sufficiently regular" random variable. Here, $H$ is a separable Hilbert space and $h_1,\ldots,h_n \in H$. Let $DF \in H$ be the Malliavin derivative of $F$ defined by $$ DF = \sum_{i=1}^n \partial_i f(W(h_1),\ldots,W(h_n)) h_i. $$ It is well-known (see Proposition 6.3 of David Nualart's "Malliavin Calculus" course https://bgsmath.cat/wp-content/uploads/2019/02/Course.pdf) that the density of $F$ is given by $$ p_F(t) = \mathbb E\left[1_{\{F \ge t\}}\delta\left(\frac{DF}{\|DF\|_H^2}\right)\right], $$ where $\delta$ is the Skorohod integral operator (aka Gaussian divergence operator), the dual of the Malliavin derivative operator $D$. In fact, one can further show that (see Proposition 6.1 of this Master's thesis http://diposit.ub.edu/dspace/bitstream/2445/158898/1/158898.pdf)
$$ \delta\left(\frac{DF}{\|DF\|_H^2}\right) = -\frac{LF}{\|DF\|_H^2}+\frac{\langle DF \otimes DF,D^2 F\rangle_{H \otimes H}}{\|DF\|_H^4}, $$ where $L := -\delta D$ is the "Laplacian" operator. My point is that, in line with the introductory remark about $\alpha_1(x)$, the second term looks suspiciously like something related to some sort of "mean curvature".
