I have been reading some notes and at point the author uses a notation I am not aware of and I wanted to figure out what it was. Let $T^*M$ be the cotangent bundle of a compact manifold $M$, and consider $H:T^*M\times S^{1}\rightarrow \mathbb{R}$ to be a time-dependent Hamiltonian function. Now for the purpose of what I am reading we will wanna consider Hamiltonians with specific properties. As an example the author states that Physical Hamiltonians of the form
$$H(q,p,t)=\frac{1}{2}|T(t,q)p-A(t,q)|^{2}+V(t,q)$$ where the symmetric tensor $T^*T$ is everywhere positive, satisfy the conditions. Now what is it meant by the symmetric tensor $T^*T$ is everywhere positive, is it we consider for fixed $q$ the function $T(t,q):T_{q}^*M\rightarrow \mathbb{R}$ and then take it's differential and we want that $dT(t,q)[p,p]\geq 0$ for every $p\in T_{q}^*M$?
Any insight is appreciated, thanks in advance.
