Rigidity theorems in geometry and topology

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The proving of rigidity theorems in geometry and topology is ubiquitous not just in these areas but in the modern approach to mathematics. Two of the finest examples of such theorems are the Sphere theorem of Rauch, Klingenberg and Berger (which restricts the global topology under geometric constraints) and the Mostow Rigidity theorem (which restricts the geometry under topological constraints). To this end the PI proposes to study more general spaces and investigate rigidity phenomena for them. In previous work with R. Spatzier and B. Wilking we showed that a manifold with sectional curvature less than or equal to 1 and a conjugate point along every geodesic at t equal to pi; is (locally) isometric to a compact, rank one symmetric space. This led to a natural question about other possible rigidity phenomena along these lines: Is there rigidity if we have the same condition on conjugate points but assume sectional curvature greater than or equal to 1? There has been significant progress on this question in collaboration with Ben Schmidt and Ralf Spatzier. In joint work with C. Sormani we extended the notion of conjugate points and proved classical theorems for complete length spaces. This leads to the question of extending rigidity theorems to spaces with two sided curvature bounds in the sense of Toponogov. We have some preliminary results in this more general setting. Several other projects are proposed including the study of Riemannian submersions from compact Lie groups (continuing recent work in this area) and studying the geometry of low cohomogeneity actions on round spheres.Most of us have an intuitive understanding of the term, "curvature". Tabletops and desktops are flat while basketballs and saddles are curved. The surface of our planet is curved as well, and we know the controversy that generated in Columbus' time! The mathematical study of the curvature of objects is the purview of differential geometry. Geometers are able to quantify curvature precisely and it provides a numerical invariant that helps distinguish objects. One may distinguish objects by shape or nature. For instance, the surface of a doughnut and the surface of a coffee cup have the same nature i.e., they are both surfaces with one hole, but they are shaped differently. On the other hand, the surface of a ball (usually called a sphere) is different in shape and nature from the surface of a doughnut (usually called a torus). How can we be sure that this is always the case? One may wonder if it is possible to deform the sphere suitably so that we might end up with the torus. This is the kind of problem that differential geometry can tackle with relative ease. A sphere has positive curvature everywhere while it can be shown that no matter what shape a torus takes, it will always have zero curvature somewhere. This tells us that the two objects are somehow fundamentally different from each other. Results of this kind are of interest to geometers as well as physicists. Differential geometry is the language used to express the general theory of relativity, our best theoretical description of gravity and its effects on the universe. In general relativity, a vacuous space-time universe would be inherently flat. This idealized state is warped by the presence of masses or energy, due to their gravitational effects. Thus, gravity is the curvature in space-time, and this is used to explain phenomena such as black holes and gravitational lensing. My are of research is the study of objects in higher dimensions (just as spacetime is four dimensional) that have positive curvature and exhibit rigidity properties under suitable additional geometric restrictions. In higher dimensions, matters are far less visually apparent. One deals almost exclusively with equations and sophisticated geometrical techniques that describe the curvature of manifolds, a term that refers to objects that, roughly speaking, have no sharp edges. Manifolds of bounded size are called compact manifolds. One of the great mysteries in the study of positive curvature is the dearth of examples of (compact) manifolds that have positive curvature at every point. The techniques at hand are few and the number of known examples remains relatively small. Therefore, any light that one can shed on the structure of such manifolds would be valuable. Structure theorems such as these are what might lead us some day to determine the precise shape of the universe.