In section 1.1 under the subtitle system of classical particles with potential, the authors claim that 
"for a system of classical particles with rigid constraints, the configuration space is a Riemannian manifold X with Riemannian structure given by twice the kinetic energy."


I don't quite how the configuration space can be given by a Riemannian manifold, as it is more naturally viewed as a symplectic manifold and there appears to be no natural Riemannian structure on a symplectic manifold. Also the relation between the Riemannian structure and the kinetic energy also eludes me. The best interpretation I can think of is to impose a Riemannian structure on the cotangent bundle via Legendre transform, or the specification of a Lagrangian function. But this is not explcitly given.



Clarification of classical field theory lecture notes by P. Deligne and D. Freed
*Configuration* space is, by definition, the position space of your particles. *Phase* space, on the other hand, is the space of pairs (position, momentum). The latter has a symplectic structure; the former has a Riemannian structure.


Regarding the relationship between kinetic energy and the Riemannian structure: You will recall from your high school physics class that kinetic energy is $\frac{1}{2} mv^2$. Of course the $v^2$ is really the dot product $v \cdot v$, in other words it's $g(v,v)$, where $g$ is the Riemannian metric and $v$ is a tangent vector. The $\frac{1}{2}$ explains the "twice the kinetic energy" part.


