Topics in differential geometry

Title page......Page 1
Introduction......Page 2
1. Homology and cohomology groups of an abstract complex......Page 3
2. Product theory......Page 7
3. An example......Page 10
4. Algebra of a vector space......Page 12
5. Differentiable manifolds......Page 15
6. Multiple integrals......Page 18
1. Riemannian manifolds in Euclidean space......Page 22
2. Imbedding and rigidity problems in Euclidean space......Page 27
3. Affine connection and absolute differentiation......Page 33
4. Riemannian metric......Page 36
5. The Gauss-Bonnet formula......Page 40
1. Resume on fiber bundles......Page 44
2. Connections......Page 49
3. Local theory of connections; the curvature tensor......Page 54
4. The homomorphism $h$ and its independence of connection......Page 57
5. The homomorphism $h$ for the universal bundle......Page 61
6. The fundamental theorem......Page 66
1. Homology groups of Grassmann manifolds......Page 69
2. Differential forms in Grassmann manifolds......Page 78
3. Multiplicative properties of the cohomology ring of a Grassmann manifold......Page 87
4. Some applications......Page 93
5. Duality Theorems......Page 102