Differential Geometry of Frame Bundles
Springer Science & Business Media, 31 dic. 1988 - 234 páginas
It isn't that they can't see the solution. It is Approach your problems from the right end and begin with the answers. Then one day, that they can't see the problem perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gu!ik's The Chillese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.
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adapted arbitrary vector fields associated complete lift components consider contact structure Corollary covariant derivative deduce defined on FM Definition denote diagonal lift diffeomorphism differentiable function differentiable map direct computation equivalent fibre bundle fibred manifold field F field of frames field of type field on FM FJ\M flat FM with respect FM,gr frame bundle function f G-bundle G-structure gl(n Gl(V hence homomorphism horizontal lift horizontal vector field induced coordinates integral curve isotropy group Kahler Killing vector field Lemma Let F Let G Levi-Civita connection Lie algebra Lie group Lie subgroup linear connection linear frame linear representation matrix Moreover obtain polynomial structure principal bundle principal fibre bundle prolongation Proof proved resp Riemannian metric second order connection spacetime standard horizontal vector structural polynomial structure group structure on FM subbundle symplectic tangent bundle tensor field torsion free TPFM unique vector space vertical