Parallel Transport
In general, parallel transport along a curve with respect to a connection defines isomorphisms between the tangent spaces at the points of the curve. If the connection is a Levi-Civita connection, then these isomorphisms are orthogonal – that is, they preserve the inner products on the various tangent spaces.
Read more about this topic: Levi-Civita Connection
Other articles related to "parallel transport, parallel":
... The introduction of parallel transport, covariant derivatives and connection forms gave a more conceptual and uniform way of understanding curvature, which ... by Levi-Civita (1917) who introduced the notion of parallel transport on surfaces ... The monodromy of this equation defines parallel transport for the connection, a notion introduced in this context by Levi-Civita ...
... See also parallel transport Given a curve in the Euclidean plane and a vector at the starting point, the vector can be transported along the curve by requiring the moving vector to ... Parallel transport can always be defined along curves on a surface using only the metric on the surface ... Parallel transport along geodesics, the "straight lines" of the surface, is easy to define ...
... Parallel transport can be discretely approximated by Schild's ladder, which takes finite steps along a curve, and approximates Levi-Civita parallelogramoids by ...
... instead, it is and restricted by the requirement that the parallel transport between points and must be a linear combination of the base vectors in ... Here, expresses the parallel transport of as linear combination of the base vectors in, i.e ... For such a metric, one can construct a dual connection to make , for parallel transport using and ...
... Parallel transport of tangent vectors along a curve in the surface was the next major advance in the subject, due to Levi-Civita ... Parallel transport along geodesics, the "straight lines" of the surface, can also easily be described directly ... field v(t) along a unit speed curve c(t), with geodesic curvature kg(t), is said to be parallel along the curve if it has constant length the angle θ(t) that it makes with the velocity ...
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