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The main subject of the book is the full understanding of Weyl's formula for the volume of a tube, its roots and its implications. Another discussed approach to the study of volumes of tubes is the computation of the power series of the volume of a tube as a function of its radius. The chapter on mean values, besides its intrinsic interest, shows an interesting fact: methods which are useful for volumes are also useful for problems related with the Laplacian. Historical notes and Mathematica drawings have been added to this revised second edition.
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2nd edition

Table of Contents

1 An Introduction to Weyl's Tube Formula.- 3 The Riccati Equation for Second Fundamental Forms.- 4 The Proof of Weyl's Tube Formula.- 5 The Generalized Gauss-Bonnet Theorem.- 6 Chern Forms and Chern Numbers.- 7 The Tube Formula in the Complex Case.- 8 Comparison Theorems for Tube Volumes.- 9 Power Series Expansions for Tube Volumes.- 10 Steiner's Formula.- 11 Mean-value Theorems.- Appendix A.- A.2 Moments.- A.3 Computation of the Volume of a Geodesic Ball.- Appendix B.- Notation Index.- Name Index.


"The new book by Alfred Gray will do much to make Weyl's tube formula more accessible to modern readers. The first five chapters give a careful and thorough discussion of each step in the derivation and its application to the Gauss-Bonnet formula. Gray's pace is quite leisurely, and a gradualte student who has completed a basic differential geometry course will have little difficulty following the presentation. In the remaining chapters of the book, one can find an extension of Weyl's tube formula to complex submanifolds of complex projective space, power series expansions for tube volumes, and the 'half-tube formula' for hypersurfaces. A high point is the presentation of estimates for the volumes of tubes in ambient Riemannian manifolds whose curvature is bounded above or below." - BULLETIN OF THE AMS (Review of the First Edition)

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