## Modern Geometry— Methods and Applications: Part Ii, the Geometry and Topology of ManifoldsUp until recently, Riemannian geometry and basic topology were not included, even by departments or faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in the classical differential geometry of curves and surfaces which were given instead (and still are given in some places) have come gradually to be viewed as anachronisms. However, there has been hitherto no unanimous agreement as to exactly how such courses should be brought up to date, that is to say, which parts of modern geometry should be regarded as absolutely essential to a modern mathematical education, and what might be the appropriate level of abstractness of their exposition. The task of designing a modernized course in geometry was begun in 1971 in the mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University. The subject-matter and level of abstractness of its exposition were dictated by the view that, in addition to the geometry of curves and surfaces, the following topics are certainly useful in the various areas of application of mathematics (especially in elasticity and relativity, to name but two), and are therefore essential: the theory of tensors (including covariant differentiation of them); Riemannian curvature; geodesics and the calculus of variations (including the conservation laws and Hamiltonian formalism); the particular case of skew-symmetric tensors (i. e. |

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### Contents

I | 1 |

II | 5 |

III | 9 |

IV | 10 |

V | 15 |

VI | 19 |

VII | 20 |

VIII | 28 |

LXII | 160 |

LXIII | 164 |

LXIV | 166 |

LXVI | 185 |

LXVII | 189 |

LXVIII | 193 |

LXIX | 195 |

LXX | 198 |

IX | 31 |

X | 37 |

XI | 41 |

XII | 42 |

XIII | 46 |

XV | 49 |

XVI | 51 |

XVII | 53 |

XVIII | 55 |

XIX | 58 |

XX | 59 |

XXI | 62 |

XXII | 65 |

XXIII | 66 |

XXIV | 69 |

XXV | 74 |

XXVI | 76 |

XXVII | 77 |

XXVIII | 79 |

XXIX | 83 |

XXX | 86 |

XXXI | 90 |

XXXII | 93 |

XXXIII | 95 |

XXXIV | 99 |

XXXV | 102 |

XXXVII | 104 |

XXXVIII | 106 |

XXXIX | 108 |

XL | 110 |

XLI | 112 |

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XLIII | 118 |

XLIV | 122 |

XLV | 125 |

XLVI | 127 |

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XLIX | 135 |

L | 137 |

LI | 139 |

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LIII | 142 |

LIV | 143 |

LV | 144 |

LVI | 147 |

LVII | 148 |

LVIII | 150 |

LIX | 153 |

LX | 156 |

LXI | 157 |

LXXI | 201 |

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LXXX | 235 |

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LXXXV | 259 |

LXXXVI | 263 |

LXXXVII | 269 |

LXXXVIII | 278 |

LXXXIX | 286 |

XC | 289 |

XCI | 290 |

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XCIII | 294 |

XCIV | 297 |

XCV | 302 |

XCVI | 308 |

XCVII | 309 |

XCVIII | 312 |

XCIX | 314 |

C | 317 |

CI | 322 |

CII | 327 |

CIII | 333 |

CIV | 340 |

CV | 344 |

CVI | 347 |

CVII | 358 |

CVIII | 359 |

CIX | 369 |

CX | 374 |

CXI | 377 |

CXII | 381 |

CXIII | 385 |

CXIV | 393 |

CXV | 399 |

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423 | |

### Common terms and phrases

boundary characteristic classes circle closed co-ordinates commutator compact complex manifold connexion consider const Corollary corresponding covering map curvature curve defined definition denote determined diffeomorphic differential dimension discrete group element embedding equation Euclidean space examples Exercise fibre bundle Figure finite foliation follows framed normal bundle fundamental group G-connexion geodesic given Hamiltonian Hence homogeneous homomorphism homotopy class homotopy groups integral intersection invariant isometries isomorphic isotropy Jacobian knot Lemma Lie algebra Lie group linear map f matrix monodromy n-dimensional neighbourhood non-degenerate non-singular normal bundle obtain orbits orientation class orthogonal pair particular path point x0 Poisson bracket polynomial preimages principal fibre bundle proof region regular value Riemannian metric satisfying singular point smooth manifold smooth map solution sphere structure group subgroup submanifold subset subspace tangent space tangent vector tensor Theorem topological space torus trajectory transformation transition functions trivial vector field verify whence zero