What is the meaning of holonomy?
Noun. holonomy (plural holonomies) (differential geometry) 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.
How is holonomy calculated?
The holonomy of a (small, if on a sphere) triangle is equal to 2p minus the sum of the exterior angles or equal to the sum of the interior angles minus p. Let b1, b2, b3 be the interior angles of the triangle and a1, a2, a3 the exterior angles.
What is holonomy relation?
The holonomy of a connection is closely related to the curvature of the connection, via the Ambrose–Singer theorem. The study of Riemannian holonomy has led to a number of important developments. The holonomy was introduced by Élie Cartan (1926) in order to study and classify symmetric spaces.
What is parallel transport in differential geometry?
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. For instance, a Koszul connection in a vector bundle also allows for the parallel transport of vectors in much the same way as with a covariant derivative.
What is a flat connection?
A flat connection is one whose curvature form vanishes identically.
Is parallel transport a geodesic?
This gives an elegant geometric definition: a geodesic is a curve whose tangent vector is parallel-transported along itself.
What is linear connection?
a principal connection on the frame bundle of a manifold or the induced connection on any associated bundle — such a connection is equivalently given by a Cartan connection for the affine group of affine space, and is often called an affine connection. …
What is a flat vector?
In vector illustration the shapes and characters of a design are simplified representations, like icons, using the computer to create defined edges and precise curves. To make flat vectors into designs, designers layer geometric shapes that can range from simple to very complex.
What is parallel transport of vectors?
Parallel transport provides a way to compare a vector in one tangent. plane to a vector in another, by moving the vector along a curve without changing it.
What is parallel translation of a vector?
When we move vectors without changing their direction,it is called parallel translation of vectors. Vector remains same by doing so.
What is local holonomy and infinitesimal holonomy?
Local and infinitesimal holonomy. If π: P → M is a principal bundle, and ω is a connection in P, then the holonomy of ω can be restricted to the fibre over an open subset of M. Indeed, if U is a connected open subset of M, then ω restricts to give a connection in the bundle π −1U over U. The holonomy (resp.
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 the holonomy of a connection?
In each of these cases, the holonomy of the connection can be identified with a Lie group, the holonomy group. The holonomy of a connection is closely related to the curvature of the connection, via the Ambrose–Singer theorem .
What is a monodromy representation of the fundamental group?
This action of the fundamental group is a monodromy representation of the fundamental group. If π: P → M is a principal bundle, and ω is a connection in P, then the holonomy of ω can be restricted to the fibre over an open subset of M. Indeed, if U is a connected open subset of M, then ω restricts to give a connection in the bundle π −1U over U.