foreword
note on the second printing
preface
1 in the ashes of the ether: differential topology
  1. smooth manifolds
  2. submanifolds
  3. fiber bundles
  4. tangent vectors, bundles, and fields
  5. differential forms
2 looking for the forest-in the leaves: foliations
  1. integral curves
  2. distributions
  3. integrability conditions
  4. the frobenius theorem
  5. the frobenius theorem in terms of differential forms
  6. foliations
  7. leaf holonomy
  8. simple foliations
3 the fundamental theorem of calculus
  1. the maurer-cartan form
  2. lie algebras
  3. structural equation
  4. adjoint action
  5. the darboux derivative
  6. the fundamental theorem: local version
  7. the fundamental theorem: global version
  8. monodromy and completeness
4 shapes fantastic: klein geometries
  1. examples of planar klein geometries
  2. principal bundles: characterization and reduction
  3. klein geometries
  4. a fundamental property
  5. the tangent bundle of a klein geometry
  6. the meteor tracking problem
  7. the gauge view of klein geometries
5 shapes high fantastical: cartan geometries
  1. the base definition of cartan geometries
  2. the principal bundle hidden in a cartan geometry
  3. the bundle definition of a cartan geometry
  4. development, geometric orientation, and holonomy
  5. flat cartan geometries and uniformization
  6. cartan space forms
  7. symmetric spaces
6 riemannian geometry
  1. the model euclidean space
  2. euclidean and riemannian geometry
  3. the equivalence problem for riemannian metrics
  4. riemannian space forms
  5. subgeometry of a riemannian geometry
  6. isoparametric submanifolds
7 msbius geometry
  1. the msbius and weyl models
  2. msbius and weyl geometries
  3. equivalence problems for a conformal metric
  4. submanifolds of msbius geometry
  5. immersed curves
  6. immersed surfaces
8 projective geometry
  1. the projective model
  2. projective cartan geometries
  3. the geometry of geodesics
  4. the projective connection in a riemannian geometry
  5. a brief tour of projective geometry
a ehresmann connections
  1. the geometric origin of ehresmann connections
  2. the reductive case
  3. ehresmann connections generalize cartan connections
  4. covariant derivative
b rolling without slipping or twisting
  1. rolling maps
  2. the existence and uniqueness of rolling maps
  3. relation to levi-civita and normal connections
  4. transitivity of rolling without slipping or twisting
c classification of one-dimensional effective klein pairs
  1. classification of one-dimensional effective klein pairs
d differential operators obtained from symmetry
  1. real representations of so2(r)
  2. operators on riemannian surfaces
e characterization of principal bundles
bibliography
index