| Preface to the Second Edition |
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ix | |
| Preface |
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xi | |
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An Introduction to Weyl's Tube Formula |
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1 | (12) |
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The Formula and Its History |
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1 | (1) |
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Weyl's Formula for Low Dimensions |
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2 | (9) |
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11 | (2) |
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Fermi Coordinates and Fermi Fields |
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13 | (18) |
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Fermi Coordinates as Generalized Normal Coordinates |
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13 | (6) |
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A Review of Curvature Fundamentals |
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19 | (1) |
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20 | (5) |
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The Generalized Gauss Lemma |
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25 | (4) |
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29 | (2) |
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The Riccati Equation for Second Fundamental Forms |
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31 | (22) |
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The Second Fundamental Forms of the Tubular Hypersurfaces |
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32 | (4) |
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The Infinitesimal Change of Volume Function |
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36 | (4) |
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Volume of a Tube in Terms of Infinitesimal Change of Volume Function |
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40 | (2) |
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The Volume of a Tube in Rn in Terms of Its Second Fundamental Forms |
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42 | (2) |
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The Bishop-Gunther Inequalities |
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44 | (4) |
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48 | (3) |
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51 | (2) |
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The Proof of Weyl's Tube Formula |
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53 | (18) |
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54 | (3) |
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57 | (3) |
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60 | (2) |
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Averaging the Tube Integrand |
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62 | (5) |
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67 | (1) |
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68 | (3) |
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The Generalized Gauss-Bonnet Theorem |
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71 | (14) |
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Tubes around Tubular Hypersurfaces |
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72 | (1) |
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73 | (2) |
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75 | (3) |
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The Gauss-Bonnet Theorem for Hypersurfaces in R2n+1 |
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78 | (2) |
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The Tube Proof of the Generalized Gauss-Bonnet Theorem |
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80 | (1) |
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The History of the Gauss-Bonnet Theorem |
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81 | (2) |
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83 | (2) |
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Chern Forms and Chern Numbers |
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85 | (32) |
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The Chern Forms of a Kahler Manifold |
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86 | (6) |
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Spaces of Constant Holomorphic Sectional Curvature |
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92 | (2) |
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Locally Symmetric Spaces and Their Compatible Submanifolds |
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94 | (4) |
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Geodesic Balls in a Space Knhol(λ) |
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98 | (2) |
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Complex Projective Space CPn(λ) |
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100 | (1) |
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The Chern Forms of Knhol(λ) |
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101 | (4) |
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Kahler Submanifolds and Wirtinger's Inequality |
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105 | (3) |
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The Homology and Cohomology of Complex Projective Space CPn(λ) |
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108 | (1) |
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109 | (1) |
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Complex Hypersurfaces of Complex Projective Space CPn+1(λ) |
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110 | (2) |
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112 | (5) |
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The Tube Formula in the Complex Case |
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117 | (26) |
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Higher Order Curvature Identities |
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118 | (4) |
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Tubes about Complex Submanifolds of Cn |
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122 | (1) |
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Tubes in a Space Knhol(λ) of Constant Holomorphic Sectional Curvature |
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123 | (3) |
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126 | (7) |
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The Projective Weyl Tube Formula |
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133 | (2) |
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Tubes about Complex Hypersurfaces of Complex Projective Space CPn+1(λ) |
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135 | (3) |
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138 | (1) |
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Tubes about Totally Real Submanifolds of Knhol(λ) |
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139 | (1) |
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140 | (3) |
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Comparison Theorems for Tube Volumes |
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143 | (42) |
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Focal Points and Cut-focal Points |
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144 | (3) |
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Tubes about Submanifolds of a Space of Nonnegative or Nonpositive Curvature |
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147 | (10) |
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The Bishop-Gunther Inequalities Generalized to Tubes |
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157 | (7) |
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Tube Volume Estimates Involving Ricci Curvature |
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164 | (3) |
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Comparison Theorems for the Volumes of Tubes about Kahler Submanifolds |
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167 | (4) |
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Some Inequalities of Heintze and Kareher |
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171 | (3) |
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Gromov's Improvement of the Bishop-Gunther Inequalities |
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174 | (3) |
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Ball and Tube Comparison Theorems for Surfaces |
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177 | (3) |
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Comparison Theorems for Riemannian Manifolds |
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180 | (2) |
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182 | (3) |
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Power Series Expansions for Tube Volumes |
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185 | (24) |
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Power Series Expansions in Normal Coordinates |
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186 | (8) |
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The Power Series Expansion for the Volume VMm (r) of a Small Geodesic Ball |
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194 | (6) |
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Power Series Expansions in Fermi Coordinates |
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200 | (6) |
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206 | (3) |
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209 | (22) |
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212 | (4) |
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The Infinitesimal Change of Volume Function of a Hypersurface |
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216 | (3) |
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Hypersurfaces in Manifolds of Nonnegative or Nonpositive Curvature |
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219 | (4) |
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Steiner's Formula for a Space of Nonnegative or Nonpositive Curvature |
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223 | (2) |
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Inequalities that Generalize Steiner's Formula |
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225 | (2) |
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227 | (4) |
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231 | (16) |
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The Laplacian and the Euclidean Laplacian |
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232 | (4) |
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Relations between the Two Laplacians and Curvature |
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236 | (1) |
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The Power Expansion for the Mean-Value Mm(r, ƒ) |
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237 | (5) |
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242 | (5) |
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247 | (8) |
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A.1 The Volume of a Ball in Rn |
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247 | (2) |
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249 | (3) |
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A.3 Computation of the Volume of a Geodesic Ball |
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252 | (3) |
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255 | (18) |
| Index |
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273 | (4) |
| Notation Index |
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277 | (2) |
| Name Index |
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279 | |
Other versions by this Author
