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 Jan 28 comment What is torsion in differential geometry intuitively? To be more explicit, $R(X,Y)f = \nabla_X \nabla_Y f - \nabla_Y \nabla_X f - \nabla_{[X,Y]} f = XYf - YXf - [X,Y]f$, which is $T(X,Y)f$. In other words, a connection on $TM$ defines connections on all tensor powers $\bigotimes^r TM \otimes \bigotimes^s T^*M$, including the empty product $\underline{\mathbb{R}}$. The curvature the induced connection on $\underline{\mathbb{R}}$ is the torsion of $\nabla$. Aug 14 awarded Good Answer Mar 15 awarded Necromancer Nov 18 awarded Famous Question Feb 17 awarded Nice Question Dec 28 comment When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$? Deane: "I don't believe I've ever seen this result used for anything." If you're referencing Cartan's Theorem (the local version of Cartan-Ambrose-Hicks), then I believe it can be used to prove that any Riemannian manifold with $\nabla R = 0$ is locally isometric to a symmetric space. Sep 30 awarded Caucus Feb 24 comment Awfully sophisticated proof for simple facts And here I was thinking the standard proof was just the Integral Test for series convergence. Jan 22 awarded Nice Answer Jan 12 comment How to respond to “I was never much good at maths at school.” It's interesting. Pretty much everyone I know would agree that it's not a good thing when elementary school teachers tell their students that they didn't like math, either. And yet I can think of more than a handful of those same people (grad students and professors) who would have no problem telling their calculus classes that they don't like calculus, or that it's not "real" math somehow... Dec 12 comment “Softness” vs “rigidity” in Geometry Funny, I would've said that when doing geometry, you either fall into algebra or analysis. In fact, it seems to me that the more "rigid" your geometry is, the more likely you are to fall into one of the sides. By contrast, "softer" geometries can rely on topology. Nov 28 comment Fundamental motivation for several complex variables That's fair. In truth, I'm still rather ignorant about harmonic functions and their properties. I don't mean to suggest that reason (1) is the only reason one would care about them, but simply that it's a primary reason that I do (again, given my ignorance). Nov 28 answered Fundamental motivation for several complex variables Nov 26 awarded Civic Duty Oct 11 comment How to escape the inclination to be a universalist or: How to learn to stop worrying and do some research. "Maybe the bred desire that you mention is (or is related to) the emphasis on 'theory building'. Yes, theory building is great, but I personally see it through the lens of problem solving." Okay, but what if one is far more interested in theory building than problem solving, or if theory-building IS the lens by which one views things? Mar 22 awarded Popular Question Jun 29 comment How to respond to “I was never much good at maths at school.” "What makes some of you think the 'math' people learned in school is 'not real'?" Just for that one line I would upvote your comment ten times if I could. For me it is extremely refreshing to hear a professional mathematician question that line. Thank you for that. May 2 awarded Nice Answer Apr 5 comment How should one present curl and divergence in an undergraduate multivariable calculus class? To be pedantic, I think technically it's $\text{curl} = \sharp \circ \ast \circ d \circ \flat$, but no matter. Apr 4 awarded Necromancer