Fundamentals of Differential Geometry

Abstract

Library ofCongress Cataloging-in-Publication Data Lang,Serge,1927-Fundamentals of differential geometry / Serge Lang. p. cm.-(Graduate texts in mathematics: 191) Inc\udes bibliographical references and index.

An elementary notion of gauge equivalence

“…One hopes that something similar will hold in the Lagrangian case. In the cases of greatest interestgenerally covariant theories and theories invariant under gauge transformations of Yang-Mills type-the spoilers of a theory are contained in a group of symmetries that are localizable, in the sense that their infinitesimal generators are, roughly speaking, parameterized by arbitrary functions on V. 23 One expects [27, p. 1281] that the null vectors at Φ point along the orbits of the subgroup of localizable symmetries with good asymptotic behaviour. 24 And one further expects that this is equivalent to the claim that N Φ is the linear span of the spoiler vectors at Φ-so that [Φ] = Φ .…”

“…Indeed, it is a basic result that if v is a vector field on a manifold M and K a compact subset of M and the integral curve of v through a point x ∈ K is defined only for t smaller than some τ ∈ R, then there must be a time t 1 < τ after which the curve leaves K and does not return [23,Theorem IV.2.3]. A straightforward corollary is that if v is a complete vector field on M and v is a vector field on M that agrees with v outside of some compact set K ⊂ M then v is also complete.…”

Geometric Differences between the Burgers and the Camassa-Holm Equations

“…It is moreover a topological linear isomorphism since A k itself is invertible. The application of the inverse mapping theorem [26] in Banach spaces achieves the proof.…”

Some aspects of differential theories Respectfully dedicated to the memory of Serge Lang

“…Lang's monograph [20] treats vector fields on Banach manifolds, while Klingenberg's book [17] deals with the Hilbertian case. Notice that if dim M = ∞, then not all derivations of C ∞ (M ) can be described as above by vector fields.…”