Natural operations on differential forms

Natural Operations on Differential Forms

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Natural Operations in Differential Geometry

Differential geometry: a natural tool for describing symmetry operations.

Differential geometry provides a useful mathematical framework for describing the fundamental concepts in crystallography. The notions of point and associated vector spaces correspond to those of manifold and tangent space at a given point. A space-group operation is a one-to-one map acting on the manifold, whereas a point-group operation is a linear map acting between two tangent spaces of the...

Differential Forms on Noncommutative Spaces

This paper is intended as an introduction to noncommutative geometry for readers with some knowledge of abstract algebra and differential geometry. We show how to extend the theory of differential forms to the “noncommutative spaces” studied in noncommutative geometry. We formulate and prove the Hochschild-Kostant-Rosenberg theorem and an extension of this result involving the Connes differential.

On rational quadratic differential forms

In linear system theory, we often encounter the situation of investigating some quadratic functionals which represent Lyapunov functions, energy storage, performance measures, e.t.c. Such a quadratic functional is called a quadratic differential form (QDF) in the context of the behavioral approach. In the past works, a QDF is usually defined in terms of a polynomial matrix. The contribution of ...