Spin connection

Spin connection

The index theorem

Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018

Let s→=(e1,…,em)be a local orthonormal frame for TM. Then s→induces natural unitary frames ±s⌣forΔ(M);there are two possible lifts corresponding to the Z2 ambiguity in lifting from SO (m) to Spin (m). Let ∇s→=ws→be the connection 1-form of the Levi-Civita connection; w is a skew symmetric matrix of 1 forms and is a 1 form valued element of the Lie algebra of the special orthogonal group. Since the Lie algebras of SO and Spin coincide, we can lift w to a 1 form valued element of the Lie algebra of Spin and let Δ(w) act on Δ(M). This lifts the Levi-Civita connection to define the spin connection () ∇s∈ConU(Δ(M))by∇ss⌣=Δ(w)s⌣.

Micropolar continua as projective space of Skyrmions

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Journal Information

Published in Philosophical Magazine, 2022

Yongjo Lee

A vanishing metricity is imposed to give rise to the definition of the general affine connection and the spin connection , as a consequence of a covariantly vanishing tetrad , This general affine connection is not assumed to be symmetric in the lower indices.

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The index theorem

Micropolar continua as projective space of Skyrmions