Gauss's original definition of curvature, Euler-Poincaré characteristic ( Gauss-Bonnet theorem ) covariant derivative, parallel transport, geodesics, topological manifold, differentiable structure, tangent space, tangent bundle, cotangent space, (pseudo-)Riemannian metric, (torsion-free)connection, constant negative curvature of (and SL(2,R) action on) the Poincaré Half Plane, (Lie-Groups), tensor(field)s, Schwarzschild's solution of the Einstein equations.