# Questions tagged [ricci-curvature]

The tag has no usage guidance.

23 questions
Filter by
Sorted by
Tagged with
1 vote
0 answers
301 views

### About "residual" scalar curvature in Einstein warped product manifold

I have an Einstein warped product manifold $M=B \times_f F$ with warped metric $g=g^B+f^2g^F$, where $F$ is Ricci flat, then its scalar curvature is $S^F=0$. It is well known that the scalar curvature ...
• 242
1 vote
0 answers
82 views

### Obstruction for a manifold to admit a periodic Ricci flow

Let M be a (compact) smooth manifold. What kind of obstruction exist for M to admit a metric whose Ricci flow is a t-periodic flow?
• 292
0 votes
0 answers
100 views

### $k$-positive Ricci curvature

"Manifolds with $k$-positive Ricci curvature" by Jon Wolfson begins by defining the following intermediate curvature: Let $(M,g)$ be an $n$-dimensional Riemannian manifold. We say that $M$ ...
• 147
4 votes
1 answer
128 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
213 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: $$f_1(\vec x)=0, \\ \vdots \\ f_m(\vec x)=0,$$ where $\vec x$ are ...
• 423
8 votes
0 answers
354 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 ...
• 4,165
2 votes
0 answers
110 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,...
• 263
7 votes
1 answer
650 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 $Ric(M)$ ...
• 20.8k
4 votes
1 answer
117 views

• 1,772
3 votes
0 answers
168 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
819 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
379 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 ...
• 1,772
4 votes
2 answers
632 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 ...
• 2,419
8 votes
2 answers
613 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 ...
• 4,013