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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,
Calculus on Manifolds
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
A curve satisfying this condition is called a geodesic and the condition itself is called the geodesic equation. The interpretation of the geodesic equation is that the tangent vector keeps pointing in the same direction as defined by the affine connection.
Micropolar continua as projective space of Skyrmions
Published in Philosophical Magazine, 2022
The Riemann curvature tensor is defined in terms of the general affine connection And a torsion tensor is defined by We define a contortion tensor by a difference between the general affine connection and a metric compatible connection , also known as the Christoffel symbol The contortion satisfies the antisymmetric property . From this, a dislocation density tensor K is defined by [24–26], The vanishing Riemann curvature (11) and its related measures imply the theory is in the regime of elasticity. And a set of partial differential equations may lead to an integrability condition, which is sometimes called the compatibility condition. In [27], a universal expression of the compatibility condition is studied under the setting of the vanishing curvature tensor in three dimensions. This yields two distinct classes of compatibility conditions, one for the vanishing torsion and another for the non-vanishing torsion. The former is well known by Vallée's classical result [28]. This result states that the vanishing Riemann curvature tensor in the deformed body yields the compatibility conditions equivalent to the Saint-Venant compatibility conditions [29–34] where Λ is a matrix defined by the stretches U and its derivatives. We defined and .