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Clash Royale CLAN TAG #URR8PPP up vote 8 down vote favorite 1 Let $D$ be the unit $n$-ball (for concreteness). Let $betainOmega^1(D;R^n)$ be an $R^n$-valued one-form, having full rank (viewed as a section of $T^*Dotimes R^n$). Under what conditions on $beta$, does there exist a section $Q$ of $SO(n,R)$ (over $D$), such that $Qcircbeta$ is closed (hence exact)? The question is non-trivial for the following reason: if there exist such $Q$ and an $f:Dto R^n$, such that $df = Qcircbeta$, then $beta^Tcircbeta = df^Tcirc df$, and the latter is (up to a musical isomorphism) a flat metric on $D$, whose Riemann curvature tensor vanishes. So in a sense, I have an answer to my question. What I am looking for is a more explicit condition; in particular, I wonder whether there exists a condition that is linear in $beta$. For the curious, this question came up twice in two different contexts in the theory of elasticity. dg.differential-geometry differential-forms share | cite | im...

Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote 2 down vote favorite 1 Along the lines of Occam's razor, but I'd like to be able to use it in a sentence regarding something specific for my college essay. Here's the context: I grew up in the same house my father practiced chiropractic and acupuncture, and after many many years of trying to bend my life into anything besides either of these two professions, I was able to come to the startling realization that my true vocation had been laying under my nose the whole time, and I never gave it the time of day to seriously consider it. "Seriously considering becoming a chiropractor and acupuncturist was like sparking a wildfire in my heart, and I had never (insert better phrasing here-something along the lines of 'realized something so obvious yet so powerful in my entire life') " I don't know if there is a way t...