### The four-vertex-property and topology of surfaces with constant curvature

- Series
- Geometry Topology Seminar
- Time
- Monday, October 27, 2008 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 269
- Speaker
- Mohammad Ghomi – School of Mathematics, Georgia Tech

We prove that every metric of constant curvature on a compact 2-manifold M with boundary bdM induces (at least) four vertices, i.e., local extrema of geodesic curvature, on bdM, if, and only if, M is simply connected. Indeed, when M is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of constant curvature on M which induce only two vertices on bdM. Furthermore, we characterize the sphere as the only closed orientable Riemannian 2-manifold M which has the four-vertex-property, i.e., the boundary of every compact surface immersed in M has 4 vertices.