Foundations of Differentiable Manifolds and Lie Groups, Volume 94Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. It includes differentiable manifolds, tensors and differentiable forms. Lie groups and homogenous spaces, integration on manifolds, and in addition provides a proof of the de Rham theorem via sheaf cohomology theory, and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem. Those interested in any of the diverse areas of mathematics requiring the notion of a differentiable manifold will find this beginning graduate-level text extremely useful. |
Contents
II | 2 |
III | 5 |
IV | 8 |
V | 11 |
VI | 22 |
VII | 30 |
VIII | 34 |
IX | 41 |
XXXII | 140 |
XXXIII | 153 |
XXXIV | 157 |
XXXV | 161 |
XXXVI | 163 |
XXXVII | 173 |
XXXVIII | 176 |
XXXIX | 186 |
X | 50 |
XI | 53 |
XII | 54 |
XIII | 62 |
XIV | 69 |
XV | 73 |
XVI | 77 |
XVII | 81 |
XVIII | 82 |
XIX | 89 |
XX | 92 |
XXI | 98 |
XXII | 101 |
XXIII | 102 |
XXIV | 109 |
XXV | 110 |
XXVI | 112 |
XXVII | 117 |
XXVIII | 120 |
XXIX | 132 |
XXX | 137 |
XXXI | 138 |
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Common terms and phrases
a₁ Aut(V basis bilinear canonical isomorphisms cochain complex cochain map coefficients cohomology modules cohomology theory commutes compact const coordinate functions coordinate system Corollary defined definition denote diffeomorphism differentiable manifold differentiable singular differentiable structure differential ideal domain element End(V Er(M Euclidean space exact sequence exists exp tX follows germs Gl(n HaeR(M identity induces inner product integral manifold invariant vector fields isomorphism K-module left invariant vector lemma Let G Lie algebra Lie group Lie group G Lie subgroup manifold structure matrix morphism multiplication non-singular obtain open set oriented orthogonal p-forms p-simplex partition of unity presheaf PROOF prove real numbers Rham cohomology second countable sheaf cohomology sheaves singular cohomology slice smooth vector field subgroup of G submanifold subspace surjective tangent vector Theorem Let topology torsionless U₁ unique v₁ vector field vector space w₁ X₁ Y₁ zero