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awarded  Nice Answer
2d
awarded  Good Answer
Aug
28
comment Curvature in geometry-interpretation
The curvature tensor embodies all of the sectional curvatures (for all possible tangent 2-planes) in one object. The other curvatures are defined by averaging the sectional curvature over natural families of tangent 2-planes. As Otis says, be patient, study differential geometry diligently, and you'll eventually see how it all works.
Aug
26
comment 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 Euler-Lagrange 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 Euler-Lagrange 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
comment 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
comment 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 calculation-free 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
comment Writing papers in pre-LaTeX 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 pre-LaTeX 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
comment Invariance of torsion and curvature
You don't say how $\bar\omega$ and $\omega$ are related. And the proposition from Kobayashi-Nomizu is indeed the answer to what is probably your question.
Aug
9
comment 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 2-planes containing $v$
Aug
5
comment 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
comment A question on theorem 1.1 of Fritz John ultrahyperbolic pde
It might be helpful to state here what equation (7) is.
Aug
1
comment Symmetries of non-Riemannian curvature tensor
Do you have any thoughts about what kind of condition you're looking for?
Aug
1
comment Symmetries of non-Riemannian curvature tensor
Ben, this doesn't look like Ricci to me. The result is a skew-symmetric tensor and vanishes if the connection is the Levi-Civita connection.