Notes

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====== Second Fundamental Form ======

The second fundamental form (which I usually write II and pronounce "two", but here will write $Q$) is defined for an immersed hypersurface in a Riemannian manifold; frequently the case of interest is a geodesic sphere. 
If $\nu$ denotes the (outward) unit normal to immersed hypersurface $M^{n-1} \subset (N^n, g)$ and $\eta$ denotes its metric dual, then
$$Q(X,Y) := g(\nabla_X \nu, Y).$$
Complete $\nu$ to a frame $e_1, \ldots, e_{n-1}, e_n := \nu$ around a point of $M$ with coframe $\omega^1, \ldots, \omega^n := \eta$.
Then $\nabla_{e_i} \eta = h_{ij} \omega^j$ for some $h_{ij}$ and we accordingly define on forms that
$$Q(\alpha) = h_{ij} e(\omega^i)e^*(\omega^j) \alpha.$$
The second fundamental form is symmetric, hence diagonalizable; frequently we like to write it in this diagonal form $$Q(\alpha) = \sum_{i=1}^{n-1} \lambda_i e(\omega^i) e^*(\omega^i) \alpha.$$

===== Mean curvature =====

The mean curvature is a scalar-valued function on $M$ defined as $H = \sum h_{ii}$; that is, $H$ is the trace of $Q$.
One important place where the trace arises is in
$$Q(\mathrm{vol}) = H \mathrm{vol},$$
used in the (Stern--di Cerbo) global computation of [[monotonicity|Price monotonicity]], for example.
This relation holds applied to the volume form of $M$ or $N$, and is a consequence of the fact that after contracting along the $e_i$ direction, the only non-zero wedge product possible is along the $e_i$ direction as the volume form has all other terms remaining.
This behaviour could differ on other $(n-1)$-forms along $M$ in $N$ that are not proportional to the volume form, as for example $\omega^2 \wedge \cdots \wedge \omega^n$ would be missing the $h_{11}$ contribution.