There has been a previous discussion about concrete constructions using the Hodge theorem , Construction of Hodge decomposition
Let me try to ask here about a specific case where one is trying to write a metric fluctuation $h$ (about a background metric $g$) on a product of two 2-manifolds. Let the indiced $\mu, \nu$ be on the first manifold and $\alpha, \beta$ on the second manifold. (possibly what is said below works only for $\mathbb{H}^2 \times S^2$)
Then the $10$ components of this metric on this $4-manifold$ can apparently be parametrized in terms of $10$ scalars $A_1,..,A_{10}$ and a harmonic scalar $f$ as follows,
we first define two vector fields as,
$X_a = A_1 \partial_a f + A_2 \epsilon_{a b} \partial ^b f$
$Y_a = A_3 \partial_a f + A_4 \epsilon_{a b} \partial ^b f$
and then in terms of that the metric components are defined as,
$h_{\mu \nu} = A_5 g_{\mu \nu} f + (D_\mu X_\nu + D_\nu X_\mu - g_{\mu \nu} D^\lambda X_\lambda )$
$h_{\alpha \beta} = A_6 g_{\alpha \beta} f + (D_\alpha Y_\beta + D_\beta Y_\alpha - g_{\alpha \beta} D^\gamma Y_\gamma )$
$h_{\alpha \mu} = A_7 \partial_\alpha \partial_\mu f + A_8 \epsilon_{\mu \nu} \partial_\alpha \partial ^\nu f + A_9\epsilon_{\alpha \beta} \partial_\mu \partial ^\beta f + A_{10}\epsilon_{\alpha \beta} \epsilon_{\mu \nu}\partial^\nu \partial ^\beta f $
where the derivative w.r.t the metric $g$ is defined as, $D_a X_b = \partial_a X_b + \Gamma ^{c}_{ab} X_c$
- I would like to understand the above construction and how it might be generalizable to higher dimensions.
Roughly the following ideas have possibly gone into this,
In two dimensions the eigenfunctions for the vector and tensor can be constructed out of the eigenfunctions of the scalar field.(why?) So possibly the "h" constructed above is an harmonic symmetric transverse traceless tensor of rank 2
Hodge decomposition says that any k form on (any?) manifold can uniquely be written as a sum of a harmonic form, an exact form and a co-exact form. Apply this to a vector field on $S^2$ (the $X_a$ and the $Y_a$ above?). The vector field is a one-form and and so we can decompose it into a harmonic form, an exact form and a co-exact form. On S^2 there is no harmonic one form (why?). So it is just a sum of an exact form which is just this $\partial f$ and a co-exact form which is just this $\epsilon\partial f$.
In getting the first ``gf" term on $h_{\mu \nu}$ and $h_{\alpha \beta}$ one has possibly used the fact that the cohomology of 2 forms is the same as the cohomology of 0-forms for 2-manifolds?
But I still don't see the origin of the $( )$ terms in $h_{\mu \nu}$ and $h_{\alpha \beta}$ and the structure of $h_{\alpha \mu}$!