# On curvature-homogeneous metrics in dimension 3. "Differential Geometry in the Large (Conference in honor of Wolfgang Meyer's 80th birthday). Università degli studi Firenze. July 11, 2016

Invited Talk

Opening Lecture at the Conference: A Riemannian manifold $(M,g)$ is said to be {\it curvature-homogeneous} if it is homogeneous to second order, i.e., if, for any two points in $M$, the Riemannian curvature tensors are equivalent under some isometry of the two tangent spaces. Of course, a locally homogeneous metric is curvature-homogeneous, but the converse is not true in dimensions greater than $2$. Already in dimension $3$, there are many unanswered questions about the existence and generality of curvature homogeneous metrics, even locally. In this case, curvature-homogeneity is equivalent to having the eigenvalues of the Ricci curvature be constant, which is a system of partial differential equations on the metric. In this talk, I will review what is known about such metrics in dimension $3$, particularly the work of O. Kowalski and his collaborators during the 1990s and, more recently, Schmidt and Wolfson. I will show that, for certain values of the eigenvalues of the Ricci tensor, these partial differential equations are integrable by Darboux’ Method, which yields some surprising relations with classical subjects, such as the theory of holomorphic curves in the complex projective plane and contact curves in the boundary of the complex 2-ball.

### Service Performed By

- Bryant, Robert Phillip Griffiths Professor of Mathematics

### Date

- July 11, 2016

### Service or Event Name

- "Differential Geometry in the Large (Conference in honor of Wolfgang Meyer's 80th birthday)

### Host Organization

- Università degli studi Firenze

### Location or Venue

- Università degli studi Firenze (Florence, Italy)