Memoirs of the American Mathematical Society 1993; 69 pp; softcover Volume: 106 ISBN10: 0821825690 ISBN13: 9780821825693 List Price: US$29 Individual Members: US$17.40 Institutional Members: US$23.20 Order Code: MEMO/106/509
 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 oneterm course in Riemannian geometry, this book may be used as a textbook in graduate courses on differential and integral geometry. Readership Graduate students and mathematicians working in differential and integral geometry. Table of Contents  Introduction
 The basic integral formula for submanifolds of a Lie group
 Poincaré'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 ChernFederer kinematic formula
 Appendix: Fibre integrals and the smooth coarea formula
 References
