What is the meaning of holonomy?
Wiktionary. holonomynoun. Given a smooth closed curve C on a surface M, and picking any point P on that curve, the holonomy of C in M is the angle by which some vector turns as it is parallel transported along the curve C from point P all the way around and back to point P.
Is Riemannian geometry differential geometry?
Riemannian Geometry is a generalization of differential geometry. Differential geometry studies the geometry of curves and surfaces using Calculus and Linear Algebra. Riemannian Geometry studies smooth manifolds using a Riemannian metric.
What is quantum holonomy theory?
Our theory — we call it quantum holonomy theory — is based on an elementary algebra, that essentially encodes how stuff is moved around in a three-dimensional space.
How is Riemannian geometry different from non Euclidean geometry?
Riemannian geometry, also called elliptic geometry, one of the non-Euclidean geometries that completely rejects the validity of Euclid’s fifth postulate and modifies his second postulate. Simply stated, Euclid’s fifth postulate is: through a point not on a given line there is only one line parallel to the given line.
Why is Riemannian geometry important?
It enabled the formulation of Einstein’s general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.
How is Riemannian geometry different from non-Euclidean geometry?
What is the purpose of parallel transport?
The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of connecting the geometries of nearby points.
What is symplectic geometry used for?
There is a simple “philosophical” answer: Symplectic forms allow you to only measure two-dimensional quantities, not one-dimensional ones (you can measure area infinitesimally). That’s the basic difference with Riemannian geometry, where you can measure length infinitesimally.
What is holonomy in differential geometry?
Holonomy. In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported.
What is a holonomy?
Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy. The most common forms of holonomy are for connections possessing some kind of symmetry.
What is the difference between monodromy and holonomy?
For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features. Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy.
What is the holonomy of curved connections?
For curved connections, holonomy has nontrivial local and global features. Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy. The most common forms of holonomy are for connections possessing some kind of symmetry.