I'm trying to generalize the expression for a Lagrangian to a manifold that doesn't posses a metric.
To be more clear, when the configurations space has a metric, we write the lagrangian quadratic in velocities as: $L=g_{ij}u^iu^j$ where $g_{ij}$ is the metric tensor and $u=\dot{q}$ are the velocities. But how can I write a generic Lagrangian quadratic in the velocities where the metric tensor doesn't exist?
My final goal is to show that if we have a quadratic lagrangian and the eulero-lagrange equations coincide with the geodesic equations (for some affine connection that is not metric), then the Ricci curvature tensor is symmetric.