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 applications arbitrary associated assume basis called canonical chart complete lift components condition consider constant construction coordinates Corollary correspondence covariant curvature curve deduce defined Definition denote derivative determined diagonal lift diffeomorphism differentiable direct computation distribution easily equation equivalent Example exists expression fact field of type field on FM flat follows frame bundle function G-structure geodesic gi(n given gl(n hand hence Hermitian homomorphism horizontal lift identity induced integrable isomorphism Lemma Lie algebra Lie group linear connection locally manifold matrix metric Moreover natural Note obtain parallel particular polynomial principal bundle projection prolongation Proof Proposition proved rank recall Remark representation resp respect Riemannian satisfies second order shows structure symplectic taking tangent tangent vector tensor field Theorem torsion unique universal vector field vector space vertical