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Preface Acknowledgments 1 Basic Notions and Concepts 1.1 Basic Manifold Theory 1.2 Connections 1.3 Curvature Models in the Real Setting 1.4 Kaihler Geometry 1.5 Curvature Decompositions 1.6 Walker Structures 1.7 Metrics on the Cotangent Bundle 1.8 Self-dual Walker Metrics 1.9 Recurrent Curvature 1.10 Constant Curvature 1.11 The Spectral Geometry of the Curvature Tensor 2 The Geometry of Deformed Riemannian Extensions 2.1 Basic Notational Conventions 2.2 Examples ofAffine Osserman Ivanov-Petrova Manifolds 2.3 The Spectral Geometry of the Curvature Tensor of Affine Surfaces 2.4 Homogeneous 2-Dimensional Affine Surfaces 2.5 The Spectral Geometry of the Curvature Tensor of Deformed Riemannian Extensions 3 The Geometry of Modified Riemannian Extensions 3.1 Four-dimensional Osserman Manifolds and Models 3.2 para-KShler Manifolds of Constant para-holomorphic Sectional Curvature . 3.3 Higher-dimensional Osserman Metrics 3.4 Osserman Metrics with Non-trivial Jordan Normal Form 3.5 (Semi) para-complex Osserman Manifolds 4 (para)-Kahler-Weyl Manifolds 4.1 Notational Conventions 4.2 (para)-Kaihler-Weyl Structures ifm ¡õ 6 4.3 (para)-Kaihler-Weyl Structures ifm = 4 4.4 (para)-Kaihler-Weyl Lie Groups ifm = 4 4.5 (para)-Kaihler-Weyl Tensors if m = 4 4.6 Realizability of (para)-Kahler-Weyl Tensors if m = 4 Bibliography Authors' Biographies Index