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Download Géométrie projective by Nicolas Jacon PDF

By Nicolas Jacon

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Rund, M. Matsumoto, R. Miron and M. Anastasiei, A. Bejancu, Abate-Patrizio, D. S. Shen, P. Ingarden and M. Matsumoto. In the present chapter we made a brief introduction in the geometry of Finsler spaces in order to study the relationships between these spaces and the dual notion of Cartan spaces. In the following we will study: Finsler metrics, Cartan nonlinear connection, canonical metrical connections and their structure equations. )-metrics, Berwald spaces will be pointed out. We underline the important role which the Sasaki lift plays for almost Kählerian model of a Finsler manifold, as well as the new notion of homogeneous lift of the Finsler metrics in the framework of this theory.

2. The mapping F has the properties: 1° F is globally defined on the manifold TM. 2° F is a tensor field of (1,1) type on TM. 11) FoF=-/. Proof. 10) that defined on TM. 11). d. 1. 1. 2. 3) on TM, we obtain Consequently, the tensorial equations Wjk = 0, h ing. 3. 12), the Lie brackets U— ,7-r give an horizontal vector fields \_dx3 ox"J if and only if &ik = 0. The previous property allows to say that Rljk is the curvature tensor field of the nonlinear connection N. 14) is the torsion of the nonlinear connection N.

2° Chern-Rund connection RT (N^, Ujk, 0). BN* • These remarkable connections satisfy a commutative diagram: fir HI' obtained by means of connection transformations [113]. The properties of metrizability of those connections can be expressed by the following table: v — metrical CT(N) h - metrical = BV(N) 9 W ~2Cyi|0 9

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