A novel framework addresses open questions in the analysis of curved spaces
Euclidean geometry is concerned with flat space; in this space, the ascertainment that 'the shortest distance between two points is a line' mathematically refers to a unique line segment. In curved space, such as that created by rolling a flat piece of paper around a paper towel roll, or the Earth as a sphere, there may be more than one 'shortest curve' between any two points, such as the many longitude lines connecting the north and south poles. These surfaces are harder to study and describe than flat ones, and Riemannian geometry is used to do so. Sub-Riemannian geometry goes beyond classical Riemannian geometry, and in many cases, the latter fails to explain the former. The EU-funded GeoSub project is developing a novel framework that should help mathematicians address open questions in sub-Riemannian geometry of importance to numerous fields in mathematics, physics and engineering.
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