Eric O. Korman
Reputation
10,926
Next privilege 15,000 Rep.
Protect questions
 Apr 24 awarded Nice Answer Feb 13 awarded Nice Answer Jan 29 awarded Good Answer Dec 12 awarded Great Question Jul 20 awarded Yearling Jul 5 awarded Nice Question Jun 22 awarded Good Question Jun 8 awarded Notable Question May 13 comment Characterization of gradient vector fields @JoshBurby unfortunately I have not thought about this anymore. May 4 awarded Nice Question Apr 26 awarded Favorite Question Apr 1 awarded Enlightened Apr 1 awarded Nice Answer Mar 16 comment How does a Lie derivative generate a $U(1)$ isometry? That's correct, $V$ is tangent to the flow at any point. This is what the equation defining $\phi_t$ says. Mar 16 answered How does a Lie derivative generate a $U(1)$ isometry? Mar 16 comment Exterior derivatives involving representations No problem! Notation in this area can be very confusing. I think most authors would write $\rho_2(\eta) \wedge \omega$ instead of $\rho_2(\eta) \circ \omega$. Mar 16 comment Exterior derivatives involving representations Let me know if my edit helps. Mar 16 revised Exterior derivatives involving representations added 711 characters in body Mar 16 comment Associated bundles: isomorphism between spaces of differential forms. So on $U_\alpha$ you just have a trivial vector bundle with fiber $V$ but then the transition functions tell you how to glue (i.e. identify) these trivial vector bundles together on overlaps. To get a global section you then need the sections on each piece to be compatible via the transition functions. Mar 16 comment Associated bundles: isomorphism between spaces of differential forms. In general, a vector bundle can be specified by the data of a vector space $V$, a cover $U_\alpha$ of your manifold, and functions $h_{\alpha\beta} : U_\alpha\cap U_\beta \to GL(V)$ satisfying the cocycle condition. Under this definition of a vector bundle, a section is a collection of functions $\zeta_\alpha : U_\alpha \to V$ such that $\zeta_\alpha(x) = h_{\alpha\beta}(x) \zeta_\beta(x)$. The associated vector bundle has the same transition functions as the principal bundle, but now acting via the rep, so the transition functions are $\rho(g_{\alpha\beta})$.