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Properties of topological spaces
Differential calculus on
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1-parameter affine connection basis called canonical Chapter chart compact components Consider constant coordinate system corresponding covariant curvature defined Definition denotes derivative dF dF diffeomorphism differentiable manifold differentiable map differential equation differential form dldx dx2 a dx3 Ehresmann connection electromagnetic equivalent Example exists exterior fiber bundle formula gauge geometrical given GL(n gravitation group G Hamiltonian hence holonomy homomorphism integral curves integral manifold isomorphism Lagrangian left invariant Lemma Lie algebra Lie group matrix metric motion n-manifold neighborhood obtain open set orientation p-form parallel translation particle Pfaffian physical principal bundle principle Proof properties Proposition Remark satisfies Section solution space-time structural group subgroup submanifold symmetry symplectic tangent space tangent vector tensor field theorem theory topological space transformation TX(M unique vector field vector space yields