15
votes
Accepted
Example of non homogenous manifold with a finitely generated algebra of natural functions
No. Here's a counterexample: Let $f(r)$ satisfy the equation $f'' + 2 f^3 = 0$ with the initial conditions $f(0)=1$ and $f'(0)=0$. Then $f$ satisfies $(f')^2+f^4=1$ and is periodic with period
$$
L ...
- 108k
12
votes
Accepted
Question on Lorentzian geometry
The signature convention $(−,+,\cdots,+)$ is more commonly used in General Relativity and Lorentzian geometry because of the desire among their practicioners to make a closer parallel to Riemannian ...
- 7,817
9
votes
Accepted
Reference request: $\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \Lambda_+$
One reference is Donaldson and Sullivan's paper "Quasiconformal 4-manifolds". See Lemma 2.3. They prove a a little more. A traceless symmetric tensor is an infinitesimal change of metric ...
- 3,429
4
votes
Accepted
Geodesic flows and Killing fields
If we have a Killing tensor field $K$ of type $(0,d)$, the function $$I:SM\to \mathbb{R}, \ I(v)= K(v,\dots, v) \ \ \ \ \ \ (\ast )$$ is constant along geodesic flow. This is a well-known ...
- 4,787
4
votes
Comparison of special metrics on Riemann surfaces with the hyperbolic one
First of all, the constants $c_i$ will have to depend on the complex structure of $X$ since without prescribing a complex structure one cannot talks about dependence on a basis of the space of ...
- 12.3k
3
votes
Reference request: $\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \Lambda_+$
There is also a quick abstract proof via representation theory: $S^2_0(\mathbb{R}^4)$ is a 9-dimensional representation of $\mathrm{SO}(4)/\{\pm I_4\}\simeq \mathrm{SO}(3)\times\mathrm{SO}(3)$ and, ...
- 108k
1
vote
Existence of Kähler Metric of Bounded Geometry on the Hermitian Vector Bundle on Projective Spaces
(This is OP, and I wrote the question with my inactive account. Please excuse.)
I think I found the proof, but there are some points that I am not 100% sure. Yet I think those problems are minor.
Let $...
- 139
1
vote
Non-compact surfaces with non-negative Gauss curvature
Here is a version using a bit less group theory:
Let $S$ be a complete surface with nonnegative Gauss (equivalently, sectional) curvature. By Perelman's proof of the soul conjecture (probably there ...
- 641
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