Curvature of Riemannian manifoldsIn mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry of surfaces and other objects. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.
Tullio Levi-CivitaTullio Levi-Civita ( à Padoue, Italie – à Rome) est un mathématicien italien. Il est connu principalement pour son travail sur le calcul tensoriel et ses applications en théorie de la relativité. Il fut l'assistant de Gregorio Ricci-Curbastro, avec qui il inventa le calcul tensoriel. Ses travaux incluent aussi des articles fondamentaux en mécanique céleste (notamment sur le problème des trois corps) et l'hydrodynamique. Né à Padoue, Levi-Civita était le fils de Giacomo Levi-Civita, un avocat qui fut sénateur.
Penrose graphical notationIn mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. A diagram in the notation consists of several shapes linked together by lines. The notation widely appears in modern quantum theory, particularly in matrix product states and quantum circuits. In particular, Categorical quantum mechanics which includes ZX-calculus is a fully comprehensive reformulation of quantum theory in terms of Penrose diagrams, and is now widely used in quantum industry.
Forme différentielle de degré unEn géométrie différentielle, les formes différentielles de degré un, ou 1-formes (différentielles), sont les exemples les plus simples de formes différentielles. Une 1-forme différentielle sur un ouvert d'un espace vectoriel normé est un champ de formes linéaires c'est-à-dire une application, qui, à chaque point de l'espace, fait correspondre une forme linéaire. Plus généralement, on peut définir de telles formes linéaires sur une variété différentielle.
TeleparallelismTeleparallelism (also called teleparallel gravity), was an attempt by Albert Einstein to base a unified theory of electromagnetism and gravity on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In this theory, a spacetime is characterized by a curvature-free linear connection in conjunction with a metric tensor field, both defined in terms of a dynamical tetrad field. The crucial new idea, for Einstein, was the introduction of a tetrad field, i.e.
