I read:
The bilinear form $(Dg)^{-1}[a](Dg)^{-T}[a]$ is positive definite on the tangent space $T_a S$ uniformly in $a \in S$
where $g:S\subset \mathbb{R}^n \to T \subset \mathbb{R}^n$.
I don't understand what the bilinear form is.. and how can it be defined on the tangent space? does the map $(Dg)^{-1}[a](\cdot)$ go from $S \to T$ or am I wrong?
Your typesetting cannot be quite right. What I see is the Jacobian $Dg$ evaluated at $a.$ They then take the inverse, call it $M = Dg^{-1}.$ The object of interest is then $M M^T,$ which is a square matrix of real numbers and is symmetric. If we take some nonzero row vector $v,$ with column vector $v^T,$ we may look at the quadratic form $$ v M M^T v^T = vM \cdot vM = |vM|^2 > 0. $$ The result is nonzero when $v\neq 0$ because $M$ is nonsingular.