An Introduction to Riemann-Finsler GeometrySpringer Science & Business Media, 17 mars 2000 - 431 pages In Riemannian geometry, measurements are made with both yardsticks and protractors. These tools are represented by a family of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in principle equipped with only a family of Minkowski norms. So ardsticks are assigned but protractors are not. With such a limited tool kit, it is natural to wonder just how much geometry one can uncover and describe? It now appears that there is a reasonable answer. Finsler geometry encompasses a solid repertoire of rigidity and comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature. There is also a bewildering array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. This book focuses on the elementary but essential items among these results. Much thought has gone into making the account a teachable one. |
Table des matières
III | 1 |
IV | 2 |
V | 5 |
VII | 6 |
VIII | 9 |
IX | 14 |
X | 15 |
XI | 17 |
XCIII | 201 |
XCIV | 203 |
XCV | 206 |
XCVI | 208 |
XCVII | 210 |
XCVIII | 213 |
XCIX | 215 |
C | 223 |
XII | 18 |
XIII | 20 |
XIV | 22 |
XV | 25 |
XVI | 27 |
XVII | 28 |
XVIII | 31 |
XIX | 33 |
XX | 37 |
XXI | 44 |
XXIII | 45 |
XXIV | 46 |
XXV | 48 |
XXVI | 49 |
XXVII | 50 |
XXVIII | 52 |
XXIX | 54 |
XXX | 55 |
XXXII | 56 |
XXXIII | 58 |
XXXIV | 63 |
XXXV | 64 |
XXXVI | 67 |
XXXVII | 69 |
XXXVIII | 70 |
XXXIX | 72 |
XL | 77 |
XLI | 82 |
XLII | 83 |
XLV | 84 |
XLVI | 85 |
XLVII | 88 |
XLVIII | 92 |
XLIX | 94 |
L | 97 |
LI | 103 |
LII | 107 |
LIII | 112 |
LIV | 113 |
LVII | 121 |
LVIII | 127 |
LIX | 131 |
LX | 137 |
LXI | 140 |
LXII | 141 |
LXIV | 142 |
LXV | 144 |
LXVI | 145 |
LXVII | 147 |
LXVIII | 148 |
LXIX | 150 |
LXX | 151 |
LXXI | 153 |
LXXII | 157 |
LXXIII | 163 |
LXXV | 164 |
LXXVI | 166 |
LXXVII | 170 |
LXXVIII | 174 |
LXXIX | 175 |
LXXXII | 178 |
LXXXIII | 181 |
LXXXV | 184 |
LXXXVI | 186 |
LXXXVII | 188 |
LXXXVIII | 192 |
LXXXIX | 193 |
XC | 194 |
XCI | 196 |
XCII | 200 |
CI | 226 |
CII | 227 |
CV | 233 |
CVI | 237 |
CVII | 240 |
CVIII | 242 |
CIX | 243 |
CX | 245 |
CXI | 246 |
CXII | 253 |
CXIII | 255 |
CXIV | 258 |
CXV | 259 |
CXVII | 260 |
CXVIII | 265 |
CXIX | 268 |
CXX | 274 |
CXXI | 277 |
CXXII | 278 |
CXXIV | 281 |
CXXV | 282 |
CXXVI | 283 |
CXXIX | 285 |
CXXX | 289 |
CXXXI | 295 |
CXXXII | 300 |
CXXXIII | 303 |
CXXXIV | 306 |
CXXXVI | 308 |
CXXXVII | 311 |
CXXXVIII | 313 |
CXLI | 314 |
CXLII | 316 |
CXLIII | 322 |
CXLIV | 327 |
CXLV | 331 |
CXLVI | 332 |
CXLVII | 333 |
CXLVIII | 335 |
CXLIX | 336 |
CL | 337 |
CLI | 338 |
CLII | 343 |
CLIII | 345 |
CLIV | 347 |
CLV | 352 |
CLVI | 353 |
CLVIII | 356 |
CLX | 357 |
CLXI | 359 |
CLXII | 363 |
CLXIV | 366 |
CLXV | 368 |
CLXVI | 371 |
CLXVII | 378 |
CLXVIII | 380 |
CLXIX | 384 |
CLXX | 385 |
CLXXII | 389 |
CLXXIII | 392 |
CLXXIV | 395 |
CLXXV | 399 |
CLXXVI | 401 |
CLXXVII | 405 |
CLXXVIII | 408 |
CLXXIX | 414 |
420 | |
421 | |
429 | |
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Expressions et termes fréquents
absolutely homogeneous Aijk arc length Berwald space bundle Cartan tensor Check Chern connection Christoffel compact computation conjugate point curve cut point defined denote diffeomorphism Differential dx¹ dx² dx³ dy¹ endpoints equation equivalent Euclidean Exercises Exercise exponential map expp Finsler geometry Finsler manifold Finsler spaces Finsler structure Finslerian formula forward geodesically complete fundamental tensor Gauss lemma Gaussian curvature geodesically complete globally Hence homogeneous of degree homotopy indicatrix inequality inner product Jacobi field Landsberg Lemma linear locally Minkowskian minimizing Minkowski norm Minkowski space natural coordinates nonzero parametrization piecewise positively homogeneous proof Proposition Randers metric Randers space reference vector Ricci Riemannian manifold Riemannian metric Rjikl scalar sectional curvature Show smooth strongly convex Suppose tangent space theorem unit speed geodesic vanishes variation vector field velocity field zero