Questions tagged [ricci-curvature]
The ricci-curvature tag has no usage guidance.
20
questions
4
votes
1
answer
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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 ...
- 1,279
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,...
- 163
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 ...
- 7,331
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 ...
- 163
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?
- 305
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 ...
- 1,748
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}(...
- 1,748
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 ...
- 287
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 ...
- 123
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 ...
- 1,748
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 ...
- 2,319
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 ...
- 3,993