The kinematic formula in Riemannian homogeneous spaces / Ralph Howard.

Format:

Book

Author/Creator:

Howard, Ralph, 1950- author.

Series:

Memoirs of the American Mathematical Society ; Volume 106, Number 509.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 106, Number 509

Language:

English

Subjects (All):

Integral geometry.

Riemannian manifolds.

Physical Description:

1 online resource (82 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 1993.

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""1. Introduction""; ""2. The Basic Integral Formula for Submanifolds of a Lie Group""; ""3. Poincare's Formula in Homogeneous Spaces""; ""Appendix: Cauchy-Crofton Type Formulas and Invariant Volumes""; ""4. Integral Invariants of Submanifolds of Homogeneous Spaces, The Kinematic Formula, and the Transfer Principle""; ""Appendix: Crofton Type Kinematic Formulas""; ""5. The Second Fundamental Form of an Intersection""; ""6. Lemmas and Definitions""; ""7. Proof of the Kinematic Formula and the Transfer Principle""; ""8. Spaces of Constant Curvature""

""9. An Algebraic Characterization of the Polynomials in the Weyl Tube Formula""""10. The Weyl Tube Formula and the Chern-Federer Kinematic Formula""; ""Appendix: Fibre Integrals and the Smooth Coarea Formula""; ""References""

Notes:

"November 1993, Volume 106, Number 509 (fourth of 6 numbers)."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0086-3