Questions tagged [ricci-curvature]

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4 votes
1 answer
98 views

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 ...
1 vote
2 answers
153 views

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
8 votes
0 answers
326 views

What specifically is the gap in Aubin's argument about positive Ricci curvature that Paul Ehrlich alludes to?

In his paper [2], Paul Ehrlich write In [1], 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 ...
  • 4,093
2 votes
0 answers
93 views

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,...
6 votes
1 answer
392 views

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 $Ric(M)$ ...
  • 19.7k
4 votes
1 answer
100 views

Rigidity of the compact irreducible symmetric space

Let $(M^n,g)$ be an irreducible symmetric space of compact type. In particular, $(M^n,g)$ is an Einstein manifold with a positive Einstein constant. Is there any classification for $(M^n,g)$ if $(M^n,...
  • 577
15 votes
1 answer
778 views

Ricci curvature : beyond heat-like flows

Let me give you some context first: just a few days ago I found some intriguing references to Ricci flows in the setting of directed graphs. There are at least two versions of Ricci curvature in the ...
3 votes
0 answers
150 views

How to show the upperbound of the Ricci tensor preserved on 3 manifold

So I have more questions coming from Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature, the proof of ...
3 votes
0 answers
52 views

Ricci deformation of hyperkahler ALE orbifold

Let $(X^2,g)$ be a hyperkahler ALE orbifold surface. Consider its Ricci deformation equation: $$ \Delta h+2Rm(h)=0 $$ for $\text{div}_g h=\text{Tr}_gh=0$ and $h=O(r^{-\epsilon})$ as $r \to +\infty$. ...
  • 577
1 vote
0 answers
93 views

Einstein submanifold of Einstein manifold - References

Is there any work in which the conditions under which an Einstein manifold admits an Einstein submanifold are studied? If yes, can you give me the references?
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2 votes
0 answers
65 views

$CD(K,N)$ condition for non complete metric measure spaces

That's basically it. I would like to know if it's possible to define the ${\sf CD}(K,N)$ condition for metric measure spaces that are not necessarily complete. The references I have found on this ...
3 votes
0 answers
140 views

References and results for the eigenvalues of Ricci tensor

I am looking for references or results that gives estimates for every eigenvalue of the Ricci tensor. For example, the least eigenvalue is related to the minimum of the Ricci curvature, what can we ...
3 votes
2 answers
844 views

From Riemannian curvature to Ricci curvature in warped product manifold

Let $M=B \times_f F$ be a warped product of two pseudo-Riemannian manifolds. If $X, Y, Z \in L(B)$ and $U, V, W \in L(F)$, (with $L(B)$ and $ L(F)$ I mean the set of all horizontal and vertical lift ...
  • 305
2 votes
1 answer
236 views

Is there any Riemannian manifold of zero dimensional isometry group such that

Sorry if this question is belongs to MSE. I have no idea about it. Question: Is there any Riemannian manifold of zero dimensional isometry group which its Ricci curvature is positive (or maybe zero)...
  • 4,093
2 votes
0 answers
80 views

Joining metrics of positive Ricci curvature

Let $M$ be a smooth manifold such that there is a closed submanifold $S\subset M$ with a Riemannian metric $g_S$ given by the restriction of a Riemannian metric on $M$ satisfying $\mathrm{Ricci}_{g_S}(...
3 votes
0 answers
160 views

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 ...
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10 votes
2 answers
773 views

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 ...
10 votes
1 answer
357 views

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 ...
4 votes
2 answers
353 views

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 ...
8 votes
2 answers
584 views

Converse to Milnor's theorem on manifolds with nonnegative Ricci curvature

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 ...