Although tremendous progress has been made towards understanding relationships between curvature and topology, the classical topic of positive/nonnegative sectional curvature still suffers from an insufficient supply of examples. The PI proposes two different methods for exploring rigidity and constructing new examples of Riemannian manifolds with nonnegative and positive sectional curvature. The first method begins with the question: classify the left-invariant metrics with nonnegative curvature on each compact Lie group. Almost all known constructions of nonnegative curvature begin with a bi-invariant metric, so answering this question would help us measure the rigidity of the known constructions, and could lead to new examples of nonnegatively curved manifolds with (at least points of) positive curvature. The second method is related to past research in which the PI developed conditions under which vector bundles admit nonnegative curvature. More precisely, if a vector bundle admits nonnegative curvature, then a fundamental differential inequality relates the three curvatures which are visible at points of its soul: the curvature of the soul, the curvature of the connection in the normal bundle of the soul, and the curvature of planes orthogonal to the soul. Conversely, if a vector bundle admits structure which strictly satisfy this inequality, then the unit sphere bundle admits positive sectional curvature. The PI proposes to use this theorem to find new examples, obstructions, and rigidity theorems for metrics with nonnegative (respectively positive) curvature on vector bundles (respectively sphere bundles).
Differential Geometry provides the mathematical language for precisely describing Einstein's theory of relativity, particle physics, and high dimensional curved shapes (called manifolds). The PI's proposed work within this field involves the study of manifolds with positive curvature, which is a visually natural restriction on the way in which a manifold curves about in space. Past examples come from Lie groups, which are indispensable tools in diverse fields of mathematics, physics, cosmology, computer animation, and other disciplines in which simplification is achieved through symmetry. The search for new examples of manifolds with positive curvature has a long history, yet frustratingly few examples have been found. The PI proposes substantially new methods for constructing more examples, which could thereby lead to a better general understanding of the relationship between curvature and global shape.
