Questions tagged [curvature]
Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)
286
questions
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Upper bound on the sectional curvature of a Riemannian submersion
Consider the manifold $M := \operatorname{SO}(n) \times \mathbb{S}^{n-1}$, endowed with the product metric given by the bi-invariant metric of $\operatorname{SO}(n)$ and the round metric of $\mathbb{S}...
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counterexample for non- monotone curvature function on the Kazdan-Warner identity
Let $\mathbb{S}^n\subset \mathbb{R}^{n+1}$ be the unit standard sphere, $n\geq 2$. $K(\xi)=\xi_{n+1}+2$, where $\xi=(\xi_1,\ldots,\xi_{n+1})\in \mathbb{S}^n$. It is easy to see that $K(\xi)$ is ...
- 441
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66
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About planar curves on a manifold
I recently came upon the following situation (think of $\mathbb{R}^3$ to simplify): let $S$ be a compact smooth surface with $K>0$ everywhere and define
$$Q=\frac{\sup_{p}\lambda_{1}(p)}{\inf_{p}\...
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14
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1
answer
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Progress on Gromov's Conjecture of the bound of total Betti numbers
This question is a reference request.
Let $(M,g)$ be a Riemannian manifold of dimension $n$, and $b_i(M) = \dim H_i(M,\mathbb{R})$. Gromov proved it that there are constants $C(n)$ such that, if the ...
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0
answers
84
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Injectivity radius bound for a metric with bounded curvature on $\mathbb{R}^n$
My question is as follows:
Question: Is it true that if $g$ is a metric (need not be complete) on $\mathbb{R}^n$ such that $B_g(x_0, 1)\subset \subset \mathbb{R}^n$, and $g$ has bounded curvature on a ...
- 11
0
votes
1
answer
58
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Handling degenerate planes in pseudo-Riemannian geometry: impact on sectional curvature and comparison theorems
I've been studying Riemannian and pseudo-Riemannian manifolds and came across an intriguing point regarding the definition of sectional curvature in both geometries.
In pseudo-Riemannian geometry, for ...
- 45
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votes
0
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149
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Kähler manifold with negative sectional curvature
Goldberg's theorem states that every almost Kähler manifold of constant curvature is Kähler if and only if the curvature is zero. This seems to contradict the fact that the sectional curvature of the ...
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3
votes
1
answer
315
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Ricci flow and curvature
I am trying to read about geometric flows mainly Ricci flows. I have a question in mind, which I am not sure whether it's possible or not.
So my question is if one starts with a metric that has mostly ...
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2
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0
answers
118
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Asymptotics on the number of diffeomorphism classes in the Cheeger finiteness theorem
A result of Cheeger says that, given any even dimension and any $\delta > 0$, there are only a finite number of diffeomorphism types of compact simply-connected manifolds of that dimension which ...
- 3,192
4
votes
0
answers
145
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Parallel transport of global sections and Riemannian curvature
A, perhaps, naive question from an algebraist/combinatorialist teaching differential geometry. Originally asked on math.SE but didn't receive a single comment in 3 days.
Consider a (real) smooth ...
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5
votes
1
answer
274
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Clarifying a result of Klingenberg
I am looking for some clarification on a result of Klingenberg. For context, I have a complete Riemannian manifold for which I would like to compute a lower bound on the injectivity radius at each ...
- 153
5
votes
1
answer
259
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Commutative/ symmetric second covariant derivative
Consider a smooth manifold $M$ together with an affine connection (or covariant derivative) $\nabla$ on the tangent bundle $TM$.
Is it possible to have an affine connection, possibly with non-zero ...
- 51
3
votes
1
answer
140
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Curvature calculation of $S^{2n+1}=U(n+1)/U(n)$ as a homogeneous space
I asked this question at StackExchange, but got no answer. So I am reposting it here.
I eventually want to check how the Hopf fibration
$$
S^{2n+1}\to {\mathbb C}P^n
$$
satisfies the Riemannian ...
- 141
1
vote
0
answers
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Can we bound the squared Gaussian curvature of genus three triply periodic minimal surfaces?
Assume that $\mathcal{M}$ is a balanced triply periodic minimal surface of genus 3, embedded in a flat torus $T^3=\mathbb{R}^3/\Lambda$ for a lattice $\Lambda$ with volume 1. I want to understand the ...
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0
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Size of conformal factor under uniformisation
Consider closed orientable surfaces whose metrics are hyperbolic (i.e., $K=-1$) except in a region which is a hemisphere of a unit sphere attached to the hyperbolic region along a closed geodesic (of ...
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