Weight Space Decomposition for Finite-Dimensional Representations of Compact Tori Geometry Chern-Weil Closedness Theorem Geometry Extension by a Compactly Supported Bump Function Geometry Gauss-Bonnet Theorem as Euler Class Evaluation Geometry Local Formula for the Chern Connection Geometry Zero-Counting Formula for the Euler Class Geometry Compatibility of the Standard Projective Atlas Geometry Baker--Campbell--Hausdorff Local Lie Group Law Geometry Geometry Area