Tits geometry and positive curvature

Abstract

The interest in positively curved manifolds goes back to the beginning of Riemannian geometry or even to spherical and projective geometry.Likewise, the program of Tits to provide an axiomatic description of geometries whose automorphism group is a noncompact simple algebraic or Lie group goes back to projective geometry.The presence of symmetries has played a significant role in the study of positively curved manifolds during the past two decades; see, e.g., the surveys [Gr], [Wi3] and [Z].Not only has this resulted in a number of classification type theorems, it has also lead to new insights about structural properties, see, e.g., [VZ] and [Wi2], as well as to the discovery and construction of a new example [De], [GVZ].Unlike [GWZ], our work here is not motivated by the quest for new examples.On the contrary, we wish to explore rigidity properties of special actions on positively curved manifolds whose linear counterparts by work of Dadok [Dad], Cartan (see [He]), Tits [Ti1] and Burns-Spatzier [BS] ultimately are described axiomatically via so-called compact spherical buildings.The special actions we investigate are the so-called polar actions, i.e., isometric actions for which there is an (immersed) submanifold, a so-called section, that meets all orbits orthogonally.Such actions form a particularly simple, yet very rich and interesting class of manifolds and actions closely related to the transformation group itself.

Locations

• Acta Mathematica - View - PDF
• arXiv (Cornell University) - View - PDF