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Jan
13
comment Conformally flat manifold with zero scalar
In general a product of l.c.f manifolds is not l.c.f. But if I recall correctly there is a small miracle that happens when you take the product of the sphere and the hyperbolic space (with curvature +1 and -1, and this works only with these values). I'll try to post the computation later if anybody is interested. Anyway, if I correctly understand the question, this give that $\mathbb{S}^n\times \mathbb{H}^n$ answers the question in even dimensions.
Jan
11
comment Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
Maybe you already know that, but by integration by parts your norm is equal to $\left(\int_\Omega|D^2u|^2dx\right)^{1/2}$ since $\langle \nabla u,\nabla\Delta u\rangle=-|D^2u|^2+\tfrac{1}{2}\Delta|\nabla u|^2$ where $|D^2u|^2$ is the Frobenius norm of the Hessian.
Nov
13
comment Classification of 2-dimensional Alexandrov spaces
You can get something local in the same vein : if $X$ is an alexandrov surface with curvature $\geq\kappa$, then every point of $X$ has a neighborhood isometric ta a small piece of the boundary of a convex subset in the space form of dimension 3 and curvature $\kappa$.
Oct
13
comment Derivation of an expression in the Ricci flow on surfaces
Isn't there a typo after the second equal sign ?
Oct
13
comment Closed curve whose neighborhood is as large as possible
Basically, you need $\varepsilon$ to be smaller than the injectivity radius of the normal exponential map.
Oct
12
comment Closed curve whose neighborhood is as large as possible
I think what's needed is smoothness rather than convexity to get that the area of an $\varepsilon$ neighborhood is $L\times\varepsilon$, where $L$ is the length of the curve, this is the first case of the tube formula.
Oct
7
comment The Solution to the system of linear PDEs
First : the $x$ parameter is irrelevant, it just multiplies the vector field so that it just gives a scaling in time. Second : what do you want to know ?
Oct
2
comment Calculating the Riemann Christoffel tensor for a diagonal metric
You want to compute the scalar curvature of your metric $g$ if I understand correctly. Still the question stands, what use would you have of the formula you will get for the scalar curvature ?
Oct
2
comment Calculating the Riemann Christoffel tensor for a diagonal metric
I agree with Deane, except if the metric is conformal to the euclidean one, in which case the formulas get much simpler.
Oct
2
comment Does this PDE only have the trivial solution?
Can you give some context ? You have only one scalar PDE to prescribe $m(m+1)/2$ functions, so it seems a bit strange to conclude that $h$ is zero unless you have some example/evidence/extra information in mind...
Sep
23
comment Area of square to wrap a torus
For Q1. Maybe I'm missing something but since the Nash Kuiper embedding is $C^1$ and isometric, its area is the area of the flat torus you start with, and by Nash Kuiper, every flat torus is isometrically embeddable, so I don't really see what Q1 means.
Sep
19
awarded  Custodian
Sep
19
reviewed No Action Needed Summing up costs over a Markov chain
Sep
14
comment Prerequisites for reading Gregory Perelman's work
The answers to this question are relevant.
Sep
1
comment Advanced Differential Geometry Textbook
Berger was my "night stand book" during my PhD !
Aug
30
awarded  Yearling
Aug
20
revised Was this particular case of the tube formula known before Weyl and Hotelling?
added 106 characters in body
Aug
19
revised Was this particular case of the tube formula known before Weyl and Hotelling?
edited tags
Aug
19
revised Was this particular case of the tube formula known before Weyl and Hotelling?
added 19 characters in body
Aug
19
asked Was this particular case of the tube formula known before Weyl and Hotelling?