Let $M$ be a compact manifold. Let $S^2 T^*M $ be vector bundle of all symmetric $(0,2)$ tensors and $S_+^2 T^*M$ the open subset of all the positive definite ones. Does $S_+^2 T^*M $ have bounded geometry? It is understood that the metric on it is the tensor product of the induced metric on the cotangent bundle from some metric on $M$.