New answers tagged riemannian-geometry
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A question in spin geometry in dimension 8
I have an answer when the structure is Spin and not just Spin$^c$. The constants $\frac{1}{8}$ and $\frac{1}{16}$ are indeed correct, one can check in orthonormal frame as well.
For $\theta\in i\Omega^...
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Question on gamma matrices
The following affirmative answer was not given yet by Branimir Cacic:
Yes you can also see Clifford multiplication as a section of $\mathrm{End}(S)\otimes T^∗M$.
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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, ...
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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 $...
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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 ...
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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 ...
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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
$$
L ...
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4
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Accepted
Geodesic flows and Killing fields
If we have a Killing tensor field $K$ of type $(0,d)$, the function $$I:SM\to \mathbb{R}, \ I(v)= K(v,\dots, v) \ \ \ \ \ \ (\ast )$$ is constant along geodesic flow. This is a well-known ...
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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 ...
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Morse approximation with bounded number of critical points
For any manifold $M$ (dropping the Riemannian structure) there exists no such constant $k(M)$. In fact, there exists a smooth function which cannot be approximated by functions in $\operatorname{...
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Sobolev and Poincaré inequalities on compact Riemannian manifolds
Is Theorem 2.12 of Dziuk and Elliott - Finite element methods for surface PDEs perhaps what you are looking for? See also, Lemma 2 in Bonito, Demlow, and Nochetto - Finite element methods for the ...
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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 ...
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Geodesics equation on Lie groups with left invariant metrics
Suppose $G$ is a Lie group with Lie algebra $\mathfrak g$, and fix a scalar product $\langle\cdot,\cdot\rangle$ on $\mathfrak g$.
This scalar product induces a left-invariant Riemannian tensor on $G$.
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