The tangent space of the space of all Riemannian metrics of volume 1

Question

Let $(M,g)$ be a compact Riemannian manifold and $\mathcal{M}$ be the set of all Riemannian metrics on $M$. How does one show that

$$ T_g\mathcal{M}_1 = \left\{h\in\Gamma(\mathrm{Sym}^2M)\mid\int_M\mathrm{tr}_g(h)\mathrm{vol}_g=0\right\} $$ where $\mathcal{M}_1 = \{g\in\mathcal{M}\mid\int_M\mathrm{vol}_g=1\}$? (see for example A. Besse "Einstein Manifolds", p.118)

Edit: Maybe it is easier to show that $\mathcal{M}_1\subset\mathcal{M}$ is a submanifold of codimension 1? I currently have no approach to the solution of this. But if one could show this, the statement follows easily...

Answer 1

Given any metric $g$ such that $V_g(M) = 1$, let $g_t$ be any smooth $1$-parameter family of metrics such that $$ g_0 = g\text{ and }\left.\partial_tg_t\right|_{t=0} = h.$$ Let $$ V_t = V_{g_t}(M) $$ and $$ \hat{g}_t = V_t^{-1/n}g_t. $$ It is now straightforward to show that $V_{\hat{g}_t}(M) = 1$ and $$ \left.\partial_t\hat{g}_t\right|_{t=0} = h. $$