# Questions tagged [ricci-curvature]

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### Volume of balls in 3-dimensional manifolds with nonpositive Ricci tensor

The volume of a ball in a 3-dimensional Riemannian manifold with nonpositive Ricci tensor is greater or equal to the volume of an Euclidean ball with the same radius? It should not be true, but I am ...
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1 vote
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### Ricci scalar of submanifold of $\mathbf R^n$

Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations: \begin{equation} f_1(\vec x)=0, \\ \vdots \\ f_m(\vec x)=0, \end{equation} where $\vec x$ are ...
• 353
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### What specifically is the gap in Aubin's argument about positive Ricci curvature that Paul Ehrlich alludes to?

In his paper , Paul Ehrlich write In , Aubin stated a theorem which implied as a corollary that if a manifold $M$ admits a Riemannian metric with nonnegative Ricci curvature and all Ricci ...
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### The Ricci curvature is bounded below by scalar curvature

So I have more questions coming from Dr Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature. In theorem 9.4,...
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### Relation between Ricci curvature and sectional curvature for 3-manifolds

Let $(M^n,g)$ be a smooth Riemannian manifold. It is well known that if $sec(M)\geq \kappa$ then $Ric(M)\geq (n-1)\kappa$. If I understand correctly in dimensions $n\geq 4$ a lower bound on $Ric(M)$ ...
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### Explicit KE metrics

Does there exist an explicit example of a Ricci-flat, non-flat metric on a closed manifold? Kaehler--Einstein, non-flat metric on a closed manifold (excluding metrics on homogeneous spaces and ...
• 287
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### Deforming metrics from non-negative to positive Ricci curvature

Given a closed Riemannian manifold $(M,g)$ with non-negative Ricci curvature and $dim\geq 3$, when can we deform the metric to a positive Ricci curved one? I know it's impossible in general due to ...
• 123
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### Positive Ricci curvature on fiber bundles

My advisor and I are working on Ricci curvature and an anonymous referee pointed out the following conjecture: Let $F\hookrightarrow M\stackrel{\pi}{\to}B$ be a fiber bundle from a compact manifold ...
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### Lower bounds on the Ricci curvature of Kähler submanifolds of $\mathbb{C}^n$

Say that $M$ is a smooth complex algebraic variety inside $\mathbb{C}^n$, and that $M$ has Ricci curvature bounded from below when endowed with the Kähler metric induced by the Euclidean metric of the ...
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Disclaimer : I suspect the question I am about to ask is really hard, but I just want to know the status of such questions. Thanks to Milnor, we know that the $\pi_1$ any compact manifold with ...