A complex manifold admits an almost complex structure, $J$, which satisfies
$$
{J^i}\_j{J^j}\_k=-{\delta^i}\_k, 
$$
and a Hermitian metric, $g$, which satisfies 
$$
g\_{st}{J^s}\_i{J^t}\_j=g\_{ij}. \tag{1}
$$
A hyperkahler manifold, on the other hand, admits an almost quaternionic structure which satisfies 
$$
{(J\_u)^i}\_j{(J\_v)^j}\_k=-{\delta^i}\_k{\delta}\_{uv}+\varepsilon\_{uvz}{(J\_z)^i}\_k, 
$$
where $u,v,z=1,2,3$. What is the generalization of (1) for a hyperkahler manifold? In other words,
$$\tag{2}
g\_{st}{(J\_u)^s}\_i{(J\_v)^t}\_j=?
$$
If the metric is Hermitian w.r.t. each complex structure, then the the RHS of (2) must contain $g\_{ij}\delta\_{uv}$. But are there any other terms on the RHS of (2)? How does this change when the manifold is quaternionic-Kahler instead of hyperkahler?



How does the concept of a hermitian metric generalize to a hyperkahler manifold?
Here are some small facts about Quaternionic Geometry. First let us leave metrics aside, for the moment. There are 2 analogues of complex manifolds in the quaternionic world, namely triholomorphic manifolds and quaternionic manifolds. They are very different. Triholomorphic manifolds admit a 2-sphere of complex structures. When I write complex structure, I mean integrable almost complex structure. On the other hand, quaternionic manifolds are manifolds admitting an almost quaternionic structure satisfying an integrability condition. They may not admit a 2-sphere of almost complex structures globally, though they do locally. But these almost complex structures need not be integrable.


As an example of a triholomorphic manifold, think of $\mathbb{H}^n$ for instance. An example of a quaternionic manifold would be $\mathbb{H}P^n$, which does not even admit a globally defined almost complex structure.


The metric analogues of these 2 classes of manifolds, would be hyperkähler manifolds and quaternion-kähler manifolds. Their definitions can be found in many places (wikipedia for instance, or Besse's Einstein manifolds, or books or lecture notes by say, Salamon for instance).


I hope this will point you in the right direction. Note that hyperkähler manifolds have holonomy which is a subgroup of $Sp(k)$, where the dimension of the manifold is $4k$, while the holonomy of a quaternion-kähler manifold of dimension $4k$ is a subgroup of $Sp(k)\times\_{\mathbb{Z}\_2} Sp(1)$.


Edit: I will answer your question directly, for hyperkähler manifolds, where the $J\_u$ are integrable and globally defined. If $u,v,w$ is a cyclic permutation of $1,2,3$, then:


$g\_{st}(J\_u)^s\_{\,i}(J\_v)^t\_{\,j} = -g\_{st}(J\_u)^s\_{\,i}(J\_u)^t\_{\,k}(J\_w)^k\_{\,j} = -g\_{ik}(J\_w)^k\_{\,j}$


So the general answer is:


$g\_{st}(J\_u)^s\_{\,i}(J\_v)^t\_{\,j} = g\_{ij} \delta\_{uv} - \epsilon\_{uvw}g\_{ik}(J\_w)^k\_{\,j}$


where summation over repeated indices was implicitly assumed.


Edit: the OP asked me in the comment about the case where $J\_u$ is denoted by $(J\_u)\_i^{\,s}$ say (the "transpose" convention, so to speak). We repeat the first calculation here, with this alternate convention.


If $u,v,w$ is a cyclic permutation of $1,2,3$, then:


$g\_{st}(J\_u)^{\,s}\_i(J\_v)^{\,t}\_j = g\_{st}(J\_u)^{\,s}\_i(J\_w)^{\,k}\_j(J\_u)^{\,t}\_k = g\_{ik}(J\_w)^{\,k}\_j$


So the general answer with this alternate convention is:


$g\_{st}(J\_u)^{\,s}\_i(J\_v)^{\,t}\_j = g\_{ij} \delta\_{uv} + \epsilon\_{uvw}g\_{ik}(J\_w)^{\,k}\_j$


