Titlebook: Applications of Affine and Weyl Geometry; Eduardo García-Río,Peter Gilkey,Ramón Vázquez-Lore Book 2013 Springer Nature Switzerland AG 2013

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Applications of Affine and Weyl Geometry978-3-031-02405-4Series ISSN 1938-1743 Series E-ISSN 1938-1751

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https://doi.org/10.1007/978-3-658-13852-3Let Φ be a symmetric (0, 2)-tensor field on an affine manifold (., ? ) of dimension . and let σ be the natural projection from .*. to .. We set . = . = Id and adopt the notation of Section 1.7. The modified Riemannian extension .?, Φ,. is the metric of neutral signature (., . ) on the cotangent bundle given in a coordinate free fashion by taking:

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The Geometry of Modified Riemannian Extensions,Let Φ be a symmetric (0, 2)-tensor field on an affine manifold (., ? ) of dimension . and let σ be the natural projection from .*. to .. We set . = . = Id and adopt the notation of Section 1.7. The modified Riemannian extension .?, Φ,. is the metric of neutral signature (., . ) on the cotangent bundle given in a coordinate free fashion by taking:

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Basic Notions and Concepts, is needed is presented in Section 1.1. Section 1.2 introduces the notion of a connection and the associated curvature operator on a general vector bundle. Of particular interest is the case when the bundle in question is the tangent bundle. The torsion tensor . is introduced; a connection is called