Interpolation on Symmetric Spaces via the Generalized Polar Decomposition
This work provides a general framework for interpolation on symmetric spaces, which is relevant for numerical relativity and other fields requiring geometric structure preservation.
The authors construct structure-preserving interpolation operators for functions valued in symmetric spaces by leveraging the generalized polar decomposition. They demonstrate applications to Lorentzian metrics, symmetric positive-definite matrices, and the Grassmannian, and numerically interpolate the Schwarzschild metric while preserving its Lorentzian signature.
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.