11
votes

Accepted

### Relation between Ricci curvature and sectional curvature for 3-manifolds

This is definitely false. In dimension 3 if $\lambda_1,\lambda_2,\lambda_3$ are eigenvalues of the curvature operator then Ricci curvatures of eigenvectors are $\lambda_1+\lambda_2, \lambda_1+\...

- 7,742

11
votes

Accepted

### Deforming metrics from non-negative to positive Ricci curvature

There are obstructions. Perhaps the most famous comes from the theorem that, if a compact spin manifold has a metric of positive scalar curvature, then its $\hat A$-genus must vanish.
If you take a ...

- 101k

9
votes

### Deforming metrics from non-negative to positive Ricci curvature

This is not a complete answer but would be helpful. Here are a few facts:
Theorem (T. Aubin 1970 and P. Ehrlich 1976). If the Ricci curvature of a compact Riemannian manifold is
non-negative and ...

- 4,085

7
votes

Accepted

### Ricci curvature : beyond heat-like flows

A small number of authors have considered hyperbolic versions of the standard flows, see e.g. "Wave character of metrics and hyperbolic geometric flow" by De-Xing Kong and Kefeng Liu and ...

- 2,095

6
votes

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

Such a manifold exists (with zero Ricci curvature): the automorphism group of the 3-dimensional flat manifold of Hantzsche and Wendt is finite.
See the book of Charlap: Bieberbach groups and flat ...

- 3,586

4
votes

Accepted

### Positive Ricci curvature on fiber bundles

This conjecture is already proved in
Gromoll, Detlef; Walschap, Gerard, Metric foliations and curvature, Progress in Mathematics 268. Basel: Birkhäuser (ISBN 978-3-7643-8714-3/hbk). viii, 174 p. ...

- 1,716

3
votes

### Lower bounds on the Ricci curvature of Kähler submanifolds of $\mathbb{C}^n$

I agree that what Dmitri said should work. Just add some comments that could help:
In your setting a lower bound on Ricci is equivalent to a lower bound on holomorphic sectional curvature.
Every ...

- 193

2
votes

Accepted

### Ricci scalar of submanifold of $\mathbf R^n$

This is an extended comment answering Ricci scalar of sub-manifold of $\mathbf R^n$
Assuming the $f_i$ are independent, at every point $x$ their gradients span the orthogonal complement to the tangent ...

- 32.5k

2
votes

### Lower bounds on the Ricci curvature of Kähler submanifolds of $\mathbb{C}^n$

This is not a real answer, but rather a suggestion (with details missing).
There is definitely a simple way to produce such varieties. For example, any hypersurface whose completion (at infinity) is ...

- 28.4k

2
votes

Accepted

### From Riemannian curvature to Ricci curvature in warped product manifold

Using (1), the relevant trace is the following, where $e_1, \dots, e_k$ is an orthonormal frame on $F$:
\begin{align*}
\mathrm{Ric}(V,W) &= \cdots - \sum_{i=1}^k \langle e_i, R(V,e_i)W\rangle\\
&...

- 25.2k

2
votes

### From Riemannian curvature to Ricci curvature in warped product manifold

In Besse "Einstein Manifold" in Corollary 9.105 you can find:
Here, you can guess how (9.105c), (9.105e) and (9.105d) can contribute to (9.106a) which is yours (2).

1
vote

### Rigidity of the compact irreducible symmetric space

The quite recent preprint “Rigidity of $SU(n)$-type symmetric spaces” by Batat, Hall, Murphy, and Waldron continues Koiso's study.
They proved that a bi-invariant metric on $SU(n)$ is rigid when $n$ ...

- 2,158

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