Canonical Integration on Symmetric Spaces

Abstract: Symmetric spaces are fundamental in differential geometry and harmonic analysis. Examples n-spheres and Grassmann manifolds, the space of positive definite symmetric matrices, Lie groups with a symmetric product, and elliptic and hyperbolic spaces with constant sectional curvatures.

Symmetric spaces are characterised by having an isometric symmetry in each point, giving rise to a symmetric product structure on the manifold.

We give an introduction to symmetric products and Lie triple systems, which describe their tangent spaces.

A new geometric numerical integration algorithm for differential equations evolving on symmetric spaces is discussed. The integrator is constructed from canonical operations on the symmetric space, its Lie triple system (LTS), and the exponential from the LTS to the symmetric space.

differential geometryprobability

Audience: advanced learners

Young Researchers between Geometry and Stochastic Analysis 2021