Tubes

1. Front Matter

   Pages i-xiii

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2. An Introduction to Weyl’s Tube Formula

   • Alfred Gray

   Pages 1-12

3. Fermi Coordinates and Fermi Fields

   • Alfred Gray

   Pages 13-30

4. The Riccati Equation for the Second Fundamental Forms

   • Alfred Gray

   Pages 31-52

5. The Proof of Weyl’s Tube Formula

   • Alfred Gray

   Pages 53-70

6. The Generalized Gauss-Bonnet Theorem

   • Alfred Gray

   Pages 71-83

7. Chern Forms and Chern Numbers

   • Alfred Gray

   Pages 85-115

8. The Tube Formula in the Complex Case

   • Alfred Gray

   Pages 117-142

9. Comparison Theorems for Tube Volumes

   • Alfred Gray

   Pages 143-183

10. Power Series Expansions for Tube Volumes

    • Alfred Gray

    Pages 185-208

11. Steiner’s Formula

    • Alfred Gray

    Pages 209-229

12. Mean-value Theorems

    • Alfred Gray

    Pages 231-245

13. Back Matter

    Pages 247-280

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In July 1998, I received an e-mail from Alfred Gray, telling me: " . . . I am in Bilbao and working on the second edition of Tubes . . . Tentatively, the new features of the book are: 1. Footnotes containing biographical information and portraits 2. A new chapter on mean-value theorems 3. A new appendix on plotting tubes " That September he spent a week in Valencia, participating in a workshop on Differential Geometry and its Applications. Here he gave me a copy of the last version of Tubes. It could be considered a final version. There was only one point that we thought needed to be considered again, namely the possible completion of the material in Section 8. 8 on comparison theorems of surfaces with the now well-known results in arbitrary dimensions. But only one month later the sad and shocking news arrived from Bilbao: Alfred had passed away. I was subsequently charged with the task of preparing the final revision of the book for the publishers, although some special circumstances prevented me from finishing the task earlier. The book appears essentially as Alred Gray left it in September 1998. The only changes I carried out were the addition of Section 8. 9 (representing the discus­ sion we had), the inclusion of some new results on harmonic spaces, the structure of Hopf hypersurfaces in complex projective spaces and the conjecture about the volume of geodesic balls.

• Differential geometry
• Volume
• curvature
• manifold
• projective geometry

"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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