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Propagation and Energy Transfer
Published in James D. Taylor, Introduction to Ultra-Wideband Radar Systems, 2020
Robert Roussel-Dupré, Terence W. Barrett
An affine connection defines parallel transport or movement over a surface and provides a measure of curvature over a global space. An affine connection can be contrasted with a metric connection which defines changes in length and angle at local points on a surface. In 1954, Yang and Mills proposed an isotopic spin SU(2) group with an affine connection similar to the vector potential of the U(1) group of the theory of electromagnetism, but whose influence produces a rotation in internal symmetry space256 () R(θ)ψ = eiθLψ,
The index theorem
Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018
The bundles W and W0 are isomorphic on the boundary. We use parallel transport along the geodesic normal rays to extend this isomorphism from the boundary to the collar C and thereby regard ∇0W as a connection on W|C. We use a partition of unity to extend ∇0W to W over M−C. Let () P0:=P(g0,∇0W)
Calculus on Manifolds
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
As for the Lie bracket, we can find a geometrical interpretation of the torsion by considering a set of different curves in the manifold M. When considering the Lie bracket, we looked at the flows of two vector fields X and Y, but for the torsion there is a more natural choice as it is intimately related to the affine connection and therefore to the notion of parallel transport. Let us therefore start at a point p and consider the two vectors X and Y in TpM. Each of these vectors defines a unique geodesic and the geodesic equation becomes () y¨a+Γbcay˙by˙c=0,
Parallel transport of local strains
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2019
F. Milicchio, V. Varano, S. Gabriele, L. Teresi, P. E. Puddu, P. Piras
Biological applications usually analyze shapes and forms utilizing homologous anatomical landmarks, and applying the discretization to a continuum model. Shapes are embedded in curved Riemannian manifolds, and to achieve an interpolation of shapes one needs to discover the geodesic underlying the deformation paths. Riemannian manifolds are defined by a metric and a compatible connection, as detailed in do Carmo Valero (1992). Both properties permit analyses on manifolds by allowing the computations of distances between points, i.e. metrical properties, and moving on the manifold along paths called geodesics. The Parallel Transport (PT) is a connection that allows transporting geometrical data along geodesics defined on manifolds, and with a choice of connection compatible with the metric, given two vectors at a point, geometrical properties such as the angle in between and their length will be conserved after applying a PT.