By G. Hardy

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**A tour of subriemannian geometries, their geodesics and applications**

Subriemannian geometries, sometimes called Carnot-Caratheodory geometries, should be seen as limits of Riemannian geometries. additionally they come up in actual phenomenon regarding ""geometric phases"" or holonomy. Very approximately conversing, a subriemannian geometry comprises a manifold endowed with a distribution (meaning a $k$-plane box, or subbundle of the tangent bundle), known as horizontal including an internal product on that distribution.

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Press 1949 [19] K. Yano, S. Bochner: Curvature and Betti Numbers, Ann. of Math. Studies 32, Princeton Univ. T. Yau: Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci.

1. 1. 1. 1. v there are local coordinates z 1 , •• ,zn, such that __a_= X ) azv v Note that a holomorphic affine connection need not be integrable, even if M admits an affine structure: There are for instance many nonintegrable holomorphic connections on a torus (see [8]). 3 Suppose now M is compact with a holomorphic affine connection 'i/. 13 the first Chern class of M vanishes. 2 Proposition: (i) All products c~ • •• ·c~ "1 "k vanish if I:A. >.!! K - 2 (ii) If M is in addition Kahler, then all Chern classes of M vanish.

I I { z E a:n : :L z v 2 < 1 + D I:L ( z v ) 2 I 2 < 2 } • The natural embedding of D into Qn is given by i z~(1 1 z : •• :z n : 1 :r(z) v 2 ). 2 Aut D consists of all transformati ons induced by G leaving D invariant. 4 A quadric structure of Misgiven by an atlas (IVa: Ua-+ Qn)aEJ such that for all a,13EJ the transition map 1Va~V~ 1 : IVa (Ua n u 13 ) ~ IV~3 (Ua n u 13 ) is the restriction of some element of G. 5 If M carries a quadric structure, then so does every unramified covering of M. 6 If M=U/f, where Uc:Qn is an open subset and rc:AutU a subgroup without fixed points which acts properly discontinuously on U and consists only of restrictions of elements of G, then M carries a natural quadric structure.