Differential Geometry from Differential Equations

We first show how, from the general 3rd order ODE of the form , one can construct a natural Lorentzian conformal metric on the four-dimensional space . When the function satisfies a special differential condition the conformal metric possesses a conformal Killing field, , which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space ) or equivalently, on the space of solutions of the original differential equation. This construction is then generalized to the pair of differential equations, z _ss =S(z,z _s ,z _t ,z _st ,s,t) and z _tt =T(z,z _s ,z _t ,z _st ,s,t), with z _s and z _t the derivatives of z with respect to s and t. In this case, from S and T, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,z _s ,z _t ,z _st ,s,t). When the S and T satisfy differential conditions analogous to those of the 3^rd order ode, the 6-space then possesses a pair of conformal Killing fields, and which allows, via the mapping to the four-space of (z,z _s ,z _t ,z _st ) and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions on S and T, thus includes all (local) solutions of the Einstein equations. Received: 10 October 2000 / Accepted: 26 June 2001     

Differential Forms in Synthetic Differential Geometry

In this paper we establish the equivalence amongvarious kinds of pointwise forms under the presence ofappropriate symmetric connections. We show also, in thepresence of a diffusion on M, a bijection between pointwise n-forms and certain globaln-forms called localizable.     

Hopf Modules and Noncommutative Differential Geometry

We define a new algebra of noncommutative differential forms for any Hopf algebra with an invertible antipode. We prove that there is a one-to-one correspondence between anti-Yetter–Drinfeld modules, which serve as coefficients for the Hopf cyclic (co)homology, and modules which admit a flat connection with respect to our differential calculus. Thus, we show that these coefficient modules can be regarded as “flat bundles” in the sense of Connes’ noncommutative differential geometry.     

The Differential Graded Odd NilHecke Algebra

We equip the odd nilHecke algebra and its associated thick calculus category with diagrammatically local differentials. The resulting differential graded Grothendieck groups are isomorphic to two different forms of the positive part of quantum \({{\mathfrak{sl}_2}}\) at a fourth root of unity.     

The Evans Lemma of Differential Geometry

A rigorous proof is given of the Evans lemma of general relativity and differential geometry. The lemma is the subsidiary proposition leading to the Evans wave equation and proves that the eigenvalues of the d'Alembertian operator, acting on any differential form, are scalar curvatures. The Evans wave equation shows that the eigenvalues of the d'Alembertian operator, acting on any differential form, are eigenvalues of the index-contracted canonical energy momentum tensor T multiplied by the Einstein constant k. The lemma is a rigorous and general result in differential geometry, and the wave equation is a rigorous and general result for all radiated and matter fields in physics. The wave equation reduces to the main equations of physics in the appropriate limits, and unifies the four types of radiated fields thought to exist in nature: gravitational, electromagnetic, weak and strong.     

Differential and Complex Geometry of Two-Dimensional Noncommutative Tori

We analyze in detail projective modules over two-dimensional noncommutative tori and complex structures on these modules. We concentrate our attention on properties of holomorphic vectors in these modules; the theory of these vectors generalizes the theory of theta-functions. The paper is self-contained; it can be used also as an introduction to the theory of noncommutative spaces with simplest space of this kind thoroughly analyzed as a basic example.     

Differential and Twistor Geometry of the Quantum Hopf Fibration

We study a quantum version of the SU(2) Hopf fibration and its associated twistor geometry. Our quantum sphere arises as the unit sphere inside a q-deformed quaternion space . The resulting four-sphere is a quantum analogue of the quaternionic projective space . The quantum fibration is endowed with compatible non-universal differential calculi. By investigating the quantum symmetries of the fibration, we obtain the geometry of the corresponding twistor space and use it to study a system of anti-self-duality equations on , for which we find an ‘instanton’ solution coming from the natural projection defining the tautological bundle over .     

Theory of Reflective X-Ray Multilayer Structures with Graded Period and its Applications

Journal of Russian Laser Research - We report a theory of reflective X-ray multilayer structures with a graded (slowly-varying) period based on the coupled-wave method and quasiclassical asymptotic...     

Effects of Contact Geometry on Electron Transport of 1,4-Diaminobenzene

The equilibrium electron transport of 1,4-diaminobenzene sandwiched between two Au electrodes is simulated by using a first principles analysis. The results show that equilibrium conductance increases with the molecule- electrode distance decreasing, and a platform occurs at the distance varying from 1.4 A to 1.9 A, implying the insensitiveness of 1,4-diaminobenzene equilibrium conductance to molecule-electrode distance. This is helpful to understand the improved reliability and reproducibility of conductance measurements using amines.     

Effective Pyroelectric Coefficient and Polarization Offset of Compositionally Step-like Graded Ferroelectric Structures

In this paper, the effective pyroelectric coefficient and polarization offset of the compositionally step-like graded multilayer ferroelectric structures have been studied by use of the first-principles approach. It is exhibited that the dielectric gradient has a nontrivial influence on the effective pyroelectric coefficient, but has a little influence on the polarization offset; and the polarization gradient plays an important role in the abnormal hysteresis loop phenomenon of the compositionally step-like graded ferroelectric structures. Moreover, the origin of the polarization offset is explored, which can be attributed to the polarization gradient in the compositionally step-like graded structure.     

Similarity-Projection Structures: The Logical Geometry of Quantum Physics

Similarity-Projection structures abstract the numerical properties of real scalar product of rays and projections in Hilbert spaces to provide a more general framework for Quantum Physics. They are characterized by properties that possess direct physical meaning. They provide a formal framework that subsumes both classical Boolean logic concerned with sets and subsets and quantum logic concerned with Hilbert space, closed subspaces and projections. They shed light on the role of the phase factors that are central to Quantum Physics. The generalization of the notion of a self-adjoint operator to SP-structures provides a novel notion that is free of linear algebra. This work was partially supported by the Jean and Helene Alfassa fund for research in Artificial Intelligence.     

Differentials of Higher Order in Noncommutative Differential Geometry

In differential geometry, the notation dⁿ f along with the corresponding formalism has fallen into disuse since the birth of exterior calculus. However, differentials of higher-order are useful objects that can be interpreted in terms of functions on iterated tangent bundles (or in terms of jets). We generalize this notion to the case of noncommutative differential geometry. For an arbitrary algebra A, people already know how to define the differential algebra A of universal differential forms over A. We define Leibniz forms of order n (these are not forms of degree n, i.e. they are not elements of ⁿA) as particular elements of what we call the iterated frame algebra of order n, F_nA, which is itself defined as the2ⁿ tensor power of the algebra A. We give a system of generators for this iterated frame algebra and identify the A left-module of forms of order n as a particular vector subspace included in the space of universal 1-forms built over the iterated frame algebra of order n-1. We study the algebraic structure of these objects, recover the case of the commutative differential calculus of order n (Leibniz differentials) and give a few examples.     

Nonabelian Bundle Gerbes,Their Differential Geometry and Gauge Theory

Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the construction of Yang-Mills theories with 2-form gauge potential.Acknowledgement We have benefited from discussions with L. Breen, D. Husemoller, A. Alekseev, L. Castellani, J. Kalkkinen, J. Mickelsson, R. Minasian, D. Stevenson and R. Stora.