By George D. Birkhoff, Ralph Beatley

ISBN-10: 0821821016

ISBN-13: 9780821821015

A hugely prompt high-school textual content via eminent students

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Example text

E. a reduction of GL(n,lR) to G does exist if and only if the tangent bundle (TM,T,M) does admit a vector bundle atlas whose structural functions are G-valued. If this happens we shall say shortly that the tangent bundle admits a reduction to G. In terms of local sections this condition is equivalent to the existence of an open covering (U,,) of the manifold M such that for every U" there exists a system of local ~)-linear vector fields Xi, i= 1, ... n which satisfy the following conditions: " 1.

12), the vector fields DxhY and Dx vY are well defmed. e. 1. d. 14) define in fact the Otsuki connections on E associated to h and v, respectively. For the theory of the Otsuki connections we refer to T. Otsuki [122]. We conclude this section with the following theorem of existence of d-connections on E. 2. Let ~ = (E,p,M) be a vector bundle whose base manifold is para compact and N be a non-linear connection in ~. Then there exists a d-connection on E. Proof. Let (p-'(U),h) be a local chart on E such that (U,rp) is a local chart on M.

N, {3 i=l " 3. The matrix (g{3,,(x)l) belongs to G. L The frames (X i(x» are said to be adapted to the G-structure which we have considered. 2. A G-structure on M is called integrable if the natural frames (aa J" are adapted to it for every x EM. 3. Let ~ = (E,p,M) be a vector bundle with type fibre 1l" and G a Lie subgroup ofGL(m,R). One says that ~ admits a G-structure if there exists a reduction of GL(m,R) to G for the principal bundle offrames of~. The existence of a G-structure for a vector bundle (E,p,M) does not depend on the existence of a G-structure on the base manifold M.

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Basic Geometry, Third Edition by George D. Birkhoff, Ralph Beatley

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