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awarded  Nice Answer 
2d

awarded  Good Answer 
Aug
28 
comment 
Curvature in geometryinterpretation
The curvature tensor embodies all of the sectional curvatures (for all possible tangent 2planes) in one object. The other curvatures are defined by averaging the sectional curvature over natural families of tangent 2planes. As Otis says, be patient, study differential geometry diligently, and you'll eventually see how it all works. 
Aug
26 
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Fair surfaces  general mathematical theory
I'm not an expert, but I don't believe there is much of a general theory for minimizing such energies, especially ones involving derivatives of the principal curvatures (whose EulerLagrange equations are PDE's of order 6 or higher). As for energy functionals involving only the principal curvatures, one always restricts to convex surfaces, where the EulerLagrange equations are elliptic 4nd order PDE's. Even here, things are difficult, and usually special cases are studied. The main ones are the total mean curvature and the Willmore functional (I suggest googling these terms). 
Aug
24 
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Hyperfunctions supported at a point
Look at the last bullet under Examples. 
Aug
24 
comment 
Hyperfunctions supported at a point
You can also look at the wikipedia article: en.wikipedia.org/wiki/Hyperfunction 
Aug
21 
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Vector Fields in a Riemannian Manifold
Willie, I certainly wasn't knocking your "lowbrow" answer. That's the best kind, especially for anyone learning things the first time. I was just wondering about a more precise way to express the fact that we all knew in advance that the vector field had to be Killing. 
Aug
21 
comment 
Vector Fields in a Riemannian Manifold
It seems to me that there should be a calculationfree argument. Something based on the fact that the Laplacian uniquely determines the metric. So if the Laplacian commutes with the vector field, then so does the metric. 
Aug
19 
comment 
Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\u\= \Bigg(\int_{{\mathbb{R}}^N} \Delta u ^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
Integrate by parts a couple of times. 
Aug
17 
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Writing papers in preLaTeX era?
It was a Hermes typewriter and looked something like this img1.etsystatic.com/018/0/5340102/il_570xN.474729763_33b9.jpg 
Aug
17 
revised 
Writing papers in preLaTeX era?
added 24 characters in body 
Aug
16 
comment 
Elliptic operators corresponds to non vanishing vector fields
A couple of quick little comments: 1) The operator $\Delta + \epsilon X^2$ depends not only on $X$ but on the Riemannian metric used to define $\Delta$. 2) Perhaps a better thing to look at is $X^2 + \epsilon^2 \Delta$ and ask what happens as $\epsilon \rightarrow 0$? 
Aug
10 
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Invariance of torsion and curvature
You don't say how $\bar\omega$ and $\omega$ are related. And the proposition from KobayashiNomizu is indeed the answer to what is probably your question. 
Aug
9 
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Harmonic map heat flow in positive curvature
Yes, the harmonic map heat flow does smooth the metric for short time. The challenge is to find a useful lower bound of the time for which the flow exists in terms of geometric invariants of the manifolds. 
Aug
6 
awarded  Nice Answer 
Aug
5 
comment 
understanding geometry of eigen values of Ricci tensor
If you fix a vector $v$, then, up to a scalar factor, $Rc(v,v)$ is the average sectional curvature of tangent 2planes containing $v$ 
Aug
5 
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Elliptic operators corresponds to non vanishing vector fields
Ali, I assumed that you wanted the PDO to be first order and the top order term to be $X$. 
Aug
5 
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A question on theorem 1.1 of Fritz John ultrahyperbolic pde
It might be helpful to state here what equation (7) is. 
Aug
1 
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Symmetries of nonRiemannian curvature tensor
Do you have any thoughts about what kind of condition you're looking for? 
Aug
1 
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Symmetries of nonRiemannian curvature tensor
Ben, this doesn't look like Ricci to me. The result is a skewsymmetric tensor and vanishes if the connection is the LeviCivita connection. 