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.
Introduction The basic integral formula for submanifolds of a Lie group Poincare's formula in homogeneous spaces Integral invariants of submanifolds of homogeneous spaces, the kinematic formula, and the transfer principle The second fundamental form of an intersection Lemmas and definitions Proof of the kinematic formula and the transfer principle Spaces of constant curvature An algebraic characterization of the polynomials in the Weyl tube formula The Weyl tube formula and the Chern-Federer kinematic formula Appendix: Fibre integrals and the smooth coarea formula References.