# The Kinematic Formula in Riemannian Homogeneous Spaces

Share this page
*Ralph Howard*

This book shows that much of classical integral geometry can be derived from the coarea formula by some elementary techniques. Howard generalizes much of classical integral geometry from spaces of constant sectional curvature to arbitrary Riemannian homogeneous spaces. To do so, he provides a general definition of an “integral invariant” of a submanifold of the space that is sufficiently general enough to cover most cases that arise in integral geometry. Working in this generality makes it clear that the type of integral geometric formulas that hold in a space does not depend on the full group of isometries, but only on the isotropy subgroup. As a special case, integral geometric formulas that hold in Euclidean space also hold in all the simply connected spaces of constant curvature. Detailed proofs of the results and many examples are included. Requiring background of a one-term course in Riemannian geometry, this book may be used as a textbook in graduate courses on differential and integral geometry.

#### Table of Contents

# Table of Contents

## The Kinematic Formula in Riemannian Homogeneous Spaces

- Contents v6 free
- 1. Introduction 18 free
- 2. The Basic Integral Formula for Submanifolds of a Lie Group 613 free
- 3. Poincare's Formula in Homogeneous Spaces 1320
- 4. Integral Invariants of Submanifolds of Homogeneous Spaces, The Kinematic Formula, and the Transfer Principle 2835
- 5. The Second Fundamental Form of an Intersection 3441
- 6. Lemmas and Definitions 3946
- 7. Proof of the Kinematic Formula and the Transfer Principle 4552
- 8. Spaces of Constant Curvature 4956
- 9. An Algebraic Characterization of the Polynomials in the Weyl Tube Formula 5461
- 10. The Weyl Tube Formula and the Chern-Federer Kinematic Formula 6269
- Appendix: Fibre Integrals and the Smooth Coarea Formula 6673
- References 6976