# Curvature-homogeneous metrics in dimension 3. Stony Brook University Colloquium Series. Stony Brook University Mathematics Department. April 28, 2016

Invited Talk

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 $2$. (For a surface (i.e., in dimension $2$) curvature-homogeneity is equivalent to having constant Gaussian curvature, and such metrics are, of course, all classified locally and they are locally homogeneous.) 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.

### Date

• April 28, 2016

### Service or Event Name

• Stony Brook University Colloquium Series

### Host Organization

• Stony Brook University Mathematics Department

### Location or Venue

• Simons Center for Geometry and Physics, Stony Brook, NY