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Curvature: from Riemannian Geometry to Hamiltonian Systems

Speaker: 
Alessandro Scagliotti
Institution: 
SISSA
Schedule: 
Tuesday, January 14, 2020 - 16:00
Location: 
A-136
Abstract: 

In this talk we start from the notion of Riemannian curvature and we extend it to more general settings, such as Sub-Riemannian Geometry.

The basic idea is to work in the framework of Riemannian Geometry and to look for objects that contain the information of curvature and, at the same time, that could be defined in a more general setting.

We show that Jacobi vector fields along geodesics carry this information. For our purposes, we introduce a dual version of Jacobi fields, i.e. the notion of Jacobi curve.
Jacobi curves are the correct objects to start with in order to define the curvature in general settings.

References:
Agrachev, Barilari, Rizzi. Curvature: a variational approach. Memoires AMS, 2018
Agrachev. Geometry of optimal control problems and Hamiltonian systems. Lecture notes in mathematics, Springer, 2008.

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