Related papers: Pythagoras Theorem in Noncommutative Geometry
We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the…
In Euclidean geometry, the Pythagorean theorem is presented as an equation involving three squares. This paper explores how analogous expressions may be identified in spherical and hyperbolic geometries.
This survey deals with the construction of a category of spectral triples that is compatible with the Kasparov product in $KK$-theory. These notes serve as an intuitive guide to these results, avoiding the necessary technical proofs. We…
In this note we present a survey on some classical and modern approaches on Pythagorean triples. Some questions are also posed in direction of some materials under review. In particular some non commutative and operator theoretical…
New formulas for the construction of Pythagorean triples and generalizations to equations of higher powers. Application of formulas to some problems, in particular Fermat's equation with n=4.
Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational(integer)right triangles from a…
The notion of a spectral geometry on a compact metric space X is introduced. This notion serves as a discrete approximation of X motivated by the notion of a spectral triple from non-commutative geometry. A set of axioms charaterising…
The subject of this PhD thesis is noncommutative geometry - more specifically spectral triples - and how it can be generalized to semi-Riemannian manifolds generally, and Lorentzian manifolds in particular. The first half of this thesis…
The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by…
Let $G$ be a finite group. Noncommutative geometry of unital $G$-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed…
This is an attempt to model ambient space as a three-dimensional real affine space with a distinguished group of automorphisms containing the translations and acting freely and transitively on pairs consisting of a half-plane together with…
We extend the notion of triangle to "imaginary triangles" with complex valued sides and angles, and parametrize families of such triangles by plane algebraic curves. We study in detail families of triangles with two commensurable angles,…
We construct spectral triples in a sense of noncommutative differential geometry, associated with a Riemannian foliation on a compact manifold, and describe its dimension spectrum.
We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities.
This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality…
In this paper, we give a comparison version of Pythagorean Theorem to judge the lower or upper bound of the curvature of Alexandrov spaces (including Riemannian manifolds).
This is the introduction and bibliography for lecture notes of a course given at the Summer School on Noncommutative Geometry and Applications, sponsored by the European Mathematical Society, at Monsaraz and Lisboa, Portugal, September…
Starting from the formulation of pseudo-Riemannian generalisation of real spectral triples we develop the data of geometries over finite-dimensional algebras with indefinite metric and their Riemannian parts. We then discuss the Standard…
This study investigates a generalisation of the Pythagorean theorem to the lengths of conic arcs constructed symmetrically on the sides of a right triangle. It is demonstrated that the theorem remains valid whenever the conic eccentricity…
What is spectral action, how to compute it and what are the known examples? This book offers a guided tour through the mathematical habitat of noncommutative geometry \`a la Connes, deliberately unveiling the answers to these questions.…