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:
“The beginnings of altruism can be seen in children as early as the age of two. How then can we be so concerned that they count by the age of three, read by four, and walk with their hands across the overhead parallel bars by five, and not be concerned that they act with kindness to others?”
—Neil Kurshan (20th century)
“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)
“... geometry became a symbol for human relations, except that it was better, because in geometry things never go bad. If certain things occur, if certain lines meet, an angle is born. You cannot fail. It’s not going to fail; it is eternal. I found in rules of mathematics a peace and a trust that I could not place in human beings. This sublimation was total and remained total. Thus, I’m able to avoid or manipulate or process pain.”
—Louise Bourgeois (b. 1911)