Foundations of Differentiable Manifolds and Lie Groups
Foundations 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.
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1-forms Aut(K basis bilinear called canonical isomorphisms cochain complex cochain map coefficients cohomology modules cohomology theory commutes compact connected integral manifold const coordinate functions coordinate system Corollary defined denote diffeomorphism differentiable manifold differentiable singular differentiable structure differential ideal dimension domain eigenvalues element EP(M Euclidean space Ev(M exists follows germs hence identity induces inner product integral manifold invariant vector fields left invariant vector lemma Let 9 Let G Lie algebra Lie group Lie group G Lie subgroup locally Euclidean manifold structure matrix morphism multiplication non-singular obtain open neighborhood open set oriented orthogonal orthonormal partition of unity presheaf proof prove real numbers real vector space Rham cohomology second countable sheaf cohomology sheaves simplex singular cohomology slice smooth vector field subgroup of G submanifold subspace surjective tangent vector tensor product Theorem Let topology torsionless torsionless resolution unique vector field vector space zero