Interpolation on Symmetric Spaces via the Generalized Polar Decomposition

TL;DR

This paper develops structure-preserving interpolation operators for functions valued in symmetric spaces, utilizing the generalized polar decomposition to ensure geometric invariance and applicability to fields like numerical relativity.

Contribution

It introduces a novel interpolation framework for symmetric space-valued functions based on the generalized polar decomposition, enabling applications in Lorentzian metrics and Grassmannians.

Findings

Constructed interpolation operators for Lorentzian metrics, symmetric positive-definite matrices, and Grassmannians.

Provided finite element methods for numerical relativity that are frame-invariant and preserve metric signature.

Demonstrated interpolation of the Schwarzschild metric numerically.

Abstract

We construct interpolation operators for functions taking values in a symmetric space -- a smooth manifold with an inversion symmetry about every point. Key to our construction is the observation that every symmetric space can be realized as a homogeneous space whose cosets have canonical representatives by virtue of the generalized polar decomposition -- a generalization of the well-known factorization of a real nonsingular matrix into the product of a symmetric positive-definite matrix times an orthogonal matrix. By interpolating these canonical coset representatives, we derive a family of structure-preserving interpolation operators for symmetric space-valued functions. As applications, we construct interpolation operators for the space of Lorentzian metrics, the space of symmetric positive-definite matrices, and the Grassmannian. In the case of Lorentzian metrics, our interpolation operators provide a family of finite elements for numerical relativity that are frame-invariant and have signature which is guaranteed to be Lorentzian pointwise. We illustrate their potential utility by interpolating the Schwarzschild metric numerically.