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The Foundations Of Differential Geometry

Harmonic analysis on semi‐simple lie groups—i

Preface. 1. The geometry of tangent bundle. 2. Finsler spaces. 3. Lagrange spaces. 4. The geometry of cotangent bundle. 5. Hamilton spaces. 6. Cartan spaces. 7. The duality between Lagrange and Hamilton spaces. 8. Symplectic transformations of the differential geometry of T* M. 9. The dual bundle of a k-osculator bundle. 10. Linear connections on the manifold T*2M. 11. Generalized Hamilton spaces of order 2. 12. Hamilton spaces of order 2. 13. Cartan spaces of order 2. Bibliography. Index.

The Geometry of Hamilton and Lagrange Spaces

We shall call paracomplex geometry the geometry related to the algebra of paracomplex numbers [70] and, mainly, the study of the structures on differentiable manifolds called paracomplex structures. When we moreover consider a compatible neutral pseudo-Riemannian metric, we have the paraHermitian and para-Kähler structures, and their variants. This subject has been studied, since the first papers by Rashevskij [94], Libermann [69] and Patterson [90] until now, from several different points of view. Moreover, the papers related to it have appeared many times in a rather disperse way, and the different schools or authors have worked many times having no relation with one another. So, the interest for a survey on the topic is clear; and, of course, mathematicians, and in particular differential geometers, will find a lot of definitions, examples, results and references in paracomplex geometry. Furthermore, we think that the solved problems in paracomplex geometry are only a little part of the “first” questions to be studied in the theory. And this is one of the attractive features of it: to be de facto a young branch of Differential Geometry, with many significant results to be proved, as the reading of the present survey, we hope, shows. Nowadays, the subject has already applications to several topics, as: negatively curved manifolds and Anosov diffeomorphisms [47], quantizable coadjoint orbits [57], mechanics [11, 86], chronogeometry [82], elliptic geometry [32], pseudo-Riemannian space forms [38], etc. On the other hand, as is seen in certain recent works, paracomplex geometry is related to some physical problems, and we think that this work can also be useful for physicists and mathematicians working in them. In particular, since every almost para-Kählerian manifold is symplectic, those manifolds furnish a large family of symplectic manifolds. Moreover, as Kaneyuki [49] points out, the study of global geometric properties of affine symmetric spaces seems to be an interesting and important problem, since affine symmetric spaces have begun to play a central role in the theory of group representations, as is shown in [13, 30, 87]. In

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A Survey on Paracomplex Geometry

The worst case fall time is given by tf = 2.2 (2Rn•Cout + Rn•Cx) = 2.2(2*1 KΩ*8fF + 1KΩ*2fF) = 39.6ps b) The best-case rise time for a NAND gate occurs when both pMOS turn on and is given by tr = 2.2 (Rp/2) •Cout, where the total parallel resistance is Rp/2. = 2.2 * 2 KΩ/2 * 8 fF = 17.6 ps c) The worst-case rise time of a NOR gate is given by tr = 2.2 (2Rp•Cout + Rp•Cx) = 2.2(2*2 KΩ*8fF + 2KΩ*2fF) = 79.2 ps

Problem (2)

A classification of homogeneous pseudo-Riemannian structures and a characterization of each primitive class are obtained. Several examples are also given.

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Reductive homogeneous pseudo-Riemannian manifolds

We consider the sectional curvatures for metric (J4 = 1)-manifolds, and study particularly the general expression of the metric and almostproduct structure in normal coordinates for para-Kaehlerian manifolds of constant para-holomorphic sectional curvature. We also introduce models of such spaces.

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Spaces of constant para-holomorphic sectional curvature

We obtain all the homogeneous pseudo-Riemannian structures on the oscillator groups equipped with a family of left-invariant Lorentzian metrics. Moreover, in the 4-dimensional case we determine all the corresponding reductive decompositions and groups of isometries.

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Homogeneous Lorentzian structures on the oscillator groups

The connected groups acting by isometries on either the real or the complex hyperbolic spaces are determined A Lie-theoretic description of the homogeneous Riemannian, respectively Kahler, structures of linear type on these spaces is then found On both spaces, examples that are
not of linear type are given

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Pedro M. Gadea

Papers

A Survey on Paracomplex Geometry

Reductive homogeneous pseudo-Riemannian manifolds

Spaces of constant para-holomorphic sectional curvature

Homogeneous Lorentzian structures on the oscillator groups

Homogeneous Structures on Real and Complex Hyperbolic Spaces