I'm trying to understand this paper (Simons '68) and I'm a bit confused with notation. For a $p$ dimensional manifold $M$ immersed in an $n$ dimensional manifold $\bar M$ with $TM$, $NM$ respectively the tangent and normal bundle of $M$ and $SM$ "the bundle whose fibre at each point is the space of symmetric linear transformations of $T_pM \to T_pM$" he defines $$\overset {\sim} A := \, ^t\!A \circ A$$ and $$\underset \sim A := \sum_{i=1}^{n-p} \text{ad } A^{V_i} \, \text{ad } A^{V_i}$$ where $\langle ^t\!A(S),V\rangle = \langle A^V,S\rangle$, $S \in SM, \, V \in \text{Hom}(NM,SM)$, $A^{V} = - (\nabla_X V)^T$ for a tangential vector field $X$ and a normal field $V$.
What does he mean by $\text{ad } A^{V}$ ?
Is this just the adjoint, ie. the transpose?
Ultimately I'm trying to understand the equation from
Theorem 4.2.1. Let $A$ be the second fundamental form of a minimal variety. Then $A$ satisfies $$ \nabla^2 A = - A \circ \overset \sim A - \underset \sim A \circ A + \bar R(A) + \bar R'$$
and how this differential equation is linear.
Does anyone know a review of that paper? It's quite technical, and some more text wouldn't hurt. (also explaining no(ta)tions a bit more)
As always, thanks a lot for any help!