# Does the tensor bundle of a compact manifold have a bounded geometry?

Let $M$ be a compact manifold. Let $S^2 T^*M$ be the vector bundle of all symmetric $(0,2)$ tensors and $S_+^2 T^*M$ be the open subset of all 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 Riemannian metric on $M$.

• Feb 18, 2016 at 11:05
• @Kaveh May you more explain on the structure of the metric on the bundle? Feb 18, 2016 at 11:25
• I read mathoverflow.net/questions/212713/…. But I can not see the conclusion. $s_+^2T^*M$ is a subbundle of $T^*M \otimes T^*M$, so the question is that is finite tensor product of cotangent bundle has bounded geometry. Feb 18, 2016 at 11:30
• About Riemannian metric, I think the natural metric is the tensor product of a Riemannian metric on the cotangent bundle. I dont not know yet what kind of metrics exist on cotangent bundles. Feb 18, 2016 at 11:41
• in the referred post $TM$ was considered with two types of metrics: Sasaki with flat fibers - not bounded geometry, and Cheeger-Gromoll, coming from submersion $GM \to TM$ of bounded geometry. So, the choice is yours. Feb 18, 2016 at 11:51