Topics in Differential Geometry

a₁ affine connection algebra anti-symmetric b₁ base space called chain characteristic classes characteristic homomorphism coboundary cochain cocycle coefficients cohomology classes cohomology groups cohomology ring compact complex coordinate neighborhood covariant cup product cycle defined definition denote differentiable manifold differential geometry dual homomorphism dual mapping e₁ element elementary cochain En+1 En+N equations equivalent Euclidean space exterior differentiation following theorem form of degree formula G₁ G₂ Grassmann manifolds Grassmann ring group G Hence homology homotopy hypersurfaces imbedded induced integral intersection number invariant polynomials isomorphic L(G)-valued lemma Lie group linear differential forms linearly mapping f matrix Maurer-Cartan forms metric multiplication notation orientable orthogonal group P₁ Pontrjagin principal bundle projection proof properties prove quadratic differential form Riemannian manifold satisfying Schubert variety space of dimension span sphere bundle structural group subgroup tangent space tangent vectors topological universal bundle vector space zero