A Panoramic view of Riemannian Geometry by Marcel Berger

By Marcel Berger

Riemannian geometry has at the present time turn into an enormous and demanding topic. This new booklet of Marcel Berger units out to introduce readers to lots of the dwelling subject matters of the sector and produce them fast to the most effects recognized so far. those effects are acknowledged with no special proofs however the major rules concerned are defined and encouraged. this allows the reader to acquire a sweeping panoramic view of just about the whole lot of the sector. even though, because a Riemannian manifold is, even at first, a refined item, beautiful to hugely non-natural ideas, the 1st 3 chapters dedicate themselves to introducing some of the recommendations and instruments of Riemannian geometry within the such a lot usual and motivating approach, following specifically Gauss and Riemann.

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Then the map Vectl:~2l(X) >V e c t L ( X ) , E -~ E 9 (r - [n/2])~, is onto whenever r _> [n/2] (where In/2] is the integer part of n/2 and (r-In/2])11 is the trivial vector bundle of rank r - In/2]). Moreover, for r >_ n/2 the same map is bijective; see for instance [Hs], p. 99. -torsion for any q _> 1, then two vector bundles E and E' of rank r over X are isomorphic if and only if they have the same Chern classes (see [Pe]). Let us suppose now that X is a compact Riemann surface (complex curve).

Any topological vector bundle E of rank r _> 2 has the form E ~ L @ (r - 1)11, where L is a line bundle and (r - 1)11 is the trivial topological vector bundle of rank r - 1. 2) that every topological line bundle has a holomorphic structure. It follows that every topological vector bundle over a curve has a holomorphic structure. If X is a surface (compact, connected complex manifold of dimension two) the topological classification of complex vector bundles was given by Wu (see [Wu]): the isomorphism classes of complex topological vector bundles are parametrised by the set {(r, cl,c2) l r an integer > 2, cl 9 H2(X, 7/), c2 9 H4(X, 7/) }.

R ( U \ A , Q ~ ) N . . N C ( U \ A , Qm) 22 1. Vector bundles over complex manifolds It follows F(U, C2,) n . . n V(U, O,m) = f'(U, Q2) n . . n r ( U , Om) . Since U is Stein, we obtain that ql N - . L,,. Thus, for the prime ideals p~ = rad(qi) C Ox,x, we have d i m ( O x , ~ / p l ) - dim X - 1 and therefore Pi = (fi) is a principal ideal (Ox,~ is factorial). It follows that q~ = ( i f ' ) for a suitable integer ki _> 1 and J~=qiN"'Nq,, = ( . f ~ * " . f k m m) is a principal ideal. P be a torsion-free s]waf of rank r over X.

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