Parallel Transport in Riemannian Geometry
In (pseudo) Riemannian geometry, a metric connection is any connection whose parallel transport mappings preserve the metric tensor. Thus a metric connection is any connection Γ such that, for any two vectors X, Y ∈ Tγ(s)
Taking the derivative at t=0, the associated differential operator ∇ must satisfy a product rule with respect to the metric:
Read more about this topic: Parallel Transport
Famous quotes containing the words parallel, transport and/or geometry:
“We tend to be so bombarded with information, and we move so quickly, that there’s a tendency to treat everything on the surface level and process things quickly. This is antithetical to the kind of openness and perception you have to have to be receptive to poetry. ... poetry seems to exist in a parallel universe outside daily life in America.”
—Rita Dove (b. 1952)
“One may disavow and disclaim vices that surprise us, and whereto our passions transport us; but those which by long habits are rooted in a strong and ... powerful will are not subject to contradiction. Repentance is but a denying of our will, and an opposition of our fantasies.”
—Michel de Montaigne (1533–1592)
“The geometry of landscape and situation seems to create its own systems of time, the sense of a dynamic element which is cinematising the events of the canvas, translating a posture or ceremony into dynamic terms. The greatest movie of the 20th century is the Mona Lisa, just as the greatest novel is Gray’s Anatomy.”
—J.G. (James Graham)