TubesIn 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. |
Contents
III | 1 |
IV | 2 |
V | 11 |
VI | 13 |
VII | 19 |
VIII | 20 |
IX | 25 |
X | 29 |
XLVIII | 118 |
XLIX | 122 |
L | 123 |
LI | 126 |
LII | 133 |
LIII | 135 |
LIV | 138 |
LV | 139 |
XI | 31 |
XII | 32 |
XIII | 36 |
XIV | 40 |
XV | 42 |
XVI | 44 |
XVII | 48 |
XVIII | 51 |
XIX | 53 |
XX | 54 |
XXI | 57 |
XXII | 60 |
XXIII | 62 |
XXIV | 67 |
XXV | 68 |
XXVI | 71 |
XXVIII | 72 |
XXIX | 73 |
XXX | 75 |
XXXI | 78 |
XXXII | 80 |
XXXIII | 81 |
XXXIV | 83 |
XXXV | 85 |
XXXVI | 86 |
XXXVII | 92 |
XXXVIII | 94 |
XXXIX | 98 |
XL | 100 |
XLI | 101 |
XLII | 105 |
XLIII | 108 |
XLIV | 109 |
XLV | 110 |
XLVI | 112 |
XLVII | 117 |
LVI | 140 |
LVII | 143 |
LVIII | 144 |
LIX | 147 |
LX | 157 |
LXI | 164 |
LXII | 167 |
LXIII | 171 |
LXIV | 174 |
LXV | 177 |
LXVI | 180 |
LXVII | 182 |
LXVIII | 185 |
LXIX | 186 |
LXX | 194 |
LXXI | 200 |
LXXII | 206 |
LXXIII | 209 |
LXXIV | 212 |
LXXV | 216 |
LXXVI | 219 |
LXXVII | 221 |
LXXIX | 223 |
LXXX | 225 |
LXXXI | 229 |
LXXXIII | 234 |
LXXXIV | 235 |
LXXXV | 240 |
LXXXVI | 245 |
LXXXVIII | 247 |
LXXXIX | 250 |
XC | 253 |
| 271 | |
| 275 | |
| 277 | |
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Common terms and phrases
change of volume Chapter Chern numbers coefficients comparison theorems complex hypersurface complex projective space complex submanifold compute constant holomorphic sectional coordinate vector coordinate vector fields Corollary cos(t covariant derivatives curvature tensor defined definition denote differential equation dimension double form ec(p Euclidean space expm Fermi coordinates Fermi fields follows Gauss Lemma Gauss-Bonnet Theorem geodesic ball holomorphic sectional curvature inequality infinitesimal change integral Kähler form Kähler manifold Khol Math matrix mean curvature metric minfoc(P Mm(r normal coordinate vector orientable hypersurface orthonormal frame orthonormal frame field Ou(t principal curvature functions Proof Ricci curvature Riemannian manifold right-hand side second fundamental form sectional curvature sin(t sin(t√x symmetric space t₁ tangent space tangent vectors tensor field topologically embedded topologically embedded submanifold total Chern form tr S(t tubular hypersurface vector fields volume function volume(P vu(t Weyl Weyl's Tube Formula ΣΣ
Popular passages
Page 266 - Palmer, Vicente) A comparison theorem for the mean exit time from a domain in a Kahler manifold. Ann. Global Anal. Geom. 10, No.