I will give an overview of the geometry of the α-Grushin plane. The α-Grushin plane is a natural generalisation of the traditional Grushin plane, one of the simplest examples of sub-Riemannian manifolds. Its geometry can be expressed with the help of special trigonometric functions. Certain regularity and continuity properties, first established by Warner in the Riemannian case, can be used to show that the exponential map fails to be injective in any neighbourhood of a critical point. In other words, the exponential map does not behave like f(x) = x3.This result allows to define conjugate points in purely metric terms. The sub-Riemannian exponential map of the α-Grushin plane can also serve to study synthetic notions of curvature, such as the CD and the MCP conditions.

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Topic: AJS - Samuel Borza

Time: 16 dec 2022 14:00 Rome

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