Ebook: Tubes

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.

---

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.

---

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.

Content:
Front Matter....Pages i-xiii
An Introduction to Weyl’s Tube Formula....Pages 1-12
Fermi Coordinates and Fermi Fields....Pages 13-30
The Riccati Equation for the Second Fundamental Forms....Pages 31-52
The Proof of Weyl’s Tube Formula....Pages 53-70
The Generalized Gauss-Bonnet Theorem....Pages 71-83
Chern Forms and Chern Numbers....Pages 85-115
The Tube Formula in the Complex Case....Pages 117-142
Comparison Theorems for Tube Volumes....Pages 143-183
Power Series Expansions for Tube Volumes....Pages 185-208
Steiner’s Formula....Pages 209-229
Mean-value Theorems....Pages 231-245
Back Matter....Pages 247-280

---

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.

Content:
Front Matter....Pages i-xiii
An Introduction to Weyl’s Tube Formula....Pages 1-12
Fermi Coordinates and Fermi Fields....Pages 13-30
The Riccati Equation for the Second Fundamental Forms....Pages 31-52
The Proof of Weyl’s Tube Formula....Pages 53-70
The Generalized Gauss-Bonnet Theorem....Pages 71-83
Chern Forms and Chern Numbers....Pages 85-115
The Tube Formula in the Complex Case....Pages 117-142
Comparison Theorems for Tube Volumes....Pages 143-183
Power Series Expansions for Tube Volumes....Pages 185-208
Steiner’s Formula....Pages 209-229
Mean-value Theorems....Pages 231-245
Back Matter....Pages 247-280
....