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recall $F:M\to N$ induces $F^\ast : T_mM \to T_{F(m)}N$ with $(F^\ast D_v)(g) = D_v(g \circ F)$ and $F_\ast : \Omega^p(N) \to \Omega^p(M)$ with $(F_\ast\omega)_m(D_{v_1}, \cdots, D_{v_p}) = \omega_{F(m)}(F^\ast D_{v_1}, \cdots, F^\ast D_{v_p})$
$\displaystyle \int_0^2 ((\varphi_3^{-1})_\ast(\rho_3 \ \mathrm df))(e_1) \ \mathrm dx = \int_0^2 (\rho_3 \ \mathrm df)((\varphi_3^{-1})^\ast e_1) \ \mathrm dx = \int_0^2 (\rho_3 \ \mathrm df)(-e_1) \ \mathrm dx$
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Transcript for
Jan14
Jan '1915
Jan19
♦ random cohomology for quantum nerds
covariance is unacceptable