If we take how the components of a tensor on a pseudo riemannian manifold transform
$$T'^{\mu\nu}(x')=\frac{\partial x'^\mu}{\partial x^\alpha}\frac{\partial x'^\nu}{\partial x^\beta}T^{\alpha\beta}(x),$$
if $T^{\alpha\beta}(x)=0$ in a certain coordinates system, we get that the tensor vanishes only in the intersection of the charts, but this nothing tells me about what the tensor does where $U_x\cap U_{x'}=\emptyset$, with $U_x$ and $U_{x'}$ neighborhoods of the two charts.
So, as far as I know from the above relation, my tensor is null in the intersection but may be non vanishing in the rest of the chart.
Do all the charts intersect somehow and thus going through all the charts I can see it?