The answer is in general No. Take e.g. the Fubini-Study Kaehler potential
K = lnD,D = 1+Q,Q = n∑k=1zkˉzk,
with Hermitian metric
gıˉȷ = ∂ıˉ∂ˉȷK = δıˉȷD−ˉzızˉȷD2 = D−1(δıˉȷ−ˉzızˉȷD),
and inverse metric
gˉıȷ = D(δˉıȷ+ˉzˉızȷ),
and Hermitian Christoffel symbols
Γℓıȷ = ∂ıgȷˉk gˉkℓ = −ˉzıδℓȷD−ˉzȷδℓıD.
The covariant derivative of the Kaehler potential is
∇ℓK = ∂ℓK = ˉzℓD.
Now calculate the sought-for quantity
∇ı∇ȷK = ∇ı∂ȷK = ∂ı∂ȷK−Γℓıȷ ∂ℓK = ˉzıˉzȷD2 ≠ 0,
which does not vanish.
This post imported from StackExchange Mathematics at 2016-06-23 20:24 (UTC), posted by SE-user Qmechanic