Manifolds with positive sectional curvature can be characterized by the property that the sum of the three angles in any triangle is larger than 180 degrees, i.e., their geometry is similar to that of the round sphere (the surface of a ball). Global Riemannian geometry can be described as relating local invariants like curvature to global invariants; manifolds with positive or more generally non-negative curvature play an important role in this subject. Another major theme in mathematics is that of symmetry. The principal investigator has made many contributions by combining both concepts, classifying positively curved manifolds with large symmetry groups and finding new examples, one of the most difficult parts of the subject. The principal investigator also studies such objects as submanifolds of Euclidean space, a classical subject since the seminal work of Gauss.
This research project studies non-negative and positive curvature in several contexts. The work aims to find new examples of manifolds with positive curvature and to find obstructions to non-negative curvature by using concave functions arising from Jacobi fields. A frequent theme is the geometry of group actions whose orbits have small codimension, especially cohomogeneity one actions. The PI studies submanifolds of Euclidean space with nonnegative curvature and their rigidity properties. In recent years the Ricci flow has found many important applications in geometry. The PI plans to study further applications of such flows or possible modifications to the existence problem in positive curvature. The PI also studies relationships of positive or non-negative curvature with the concept of geometric formality and that of polar actions. In another project the PI studies initial value problems and existence of Einstein metrics on cohomogeneity one manifolds.