Cocurvature
Encyclopedia
In mathematics
in the branch of differential geometry, the cocurvature of a connection
on a manifold
is the obstruction to the integrability of the vertical bundle
.
is a vector-valued 2-form on M defined by
where X and Y are vector fields on M.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
in the branch of differential geometry, the cocurvature of a connection
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
on a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
is the obstruction to the integrability of the vertical bundle
Vertical bundle
The vertical bundle of a smooth fiber bundle is the subbundle of the tangent bundle that consists of all vectors which are tangent to the fibers...
.
Definition
If M is a manifold and P is a connection on M, that is a vector-valued 1-form on M which is a projection on TM such that PabPbc = Pac, then the cocurvature is a vector-valued 2-form on M defined bywhere X and Y are vector fields on M.