Smooth Manifold Definition Tangent Space Definition Diffeomorphism Definition Inverse Function Theorem Theorem #51 Functoriality of the Chern-Weil Homomorphism under Extension of Structure Group Geometry Complex Frame Bundle Is a Principal Bundle Geometry Normal Bundle of a Smooth Divisor Geometry Matrix Exponential Gives One-Parameter Subgroups of the General Linear Group Geometry Principal Bundle Reconstruction from Transition Functions Geometry Gauss-Bonnet Theorem as Euler Class Evaluation Geometry Pontryagin Classes from Chern Classes of the Complexification Geometry Jacobian of the Conjugation Map Along a Maximal Torus Geometry Geometry Area