Let $(M,h)$ be a Riemannian manifold, $G$ its isometry group and $G\times M\rightarrow M,\ (g,x)\mapsto g(x)$ the canonical action of $G$ on $M$. Let further $\mathfrak{g}$ be the Lie algebra of $G$, $X\in\mathfrak{g}$ and $\tilde{X}$ the fundamental vector field on $M$ corresponding to $X$. Finally, let $\mu$ be a probability measure on $M$ with respect to the Borel sigma algebra.
I'm now wondering if the following holds true:
$\|\tilde{X}\|_{L^2(\mu)}<\infty,$
where $\|\tilde{X}\|^2_{L^2(\mu)}:=\int_M h(\tilde{X},\tilde{X})\ d\mu$.
Thank you for any hints.
It is not true for the rotation of the hyperbolic space (and the uniform measure $\mu = r\mathrm dr\mathrm d\theta$). The metric is
$$g = \frac{4}{(1-r^2)^2}(\mathrm dr^2 + r^2 \mathrm d\theta^2).$$
Consider the rotation $z\mapsto e^{i\theta}z$. Thus $\bar X = \partial_\theta$ and $g(\bar X, \bar X) = \frac{4r^2}{(1-r^2)^2}$
Then
$$\| \bar X\|^2_{L^2(\mu)} \ge C\int_{1/2}^1 \frac{1}{(1-r)^2} dr =\infty$$