Harmonic manifolds of hypergeometric type and spherical Fourier transform

TL;DR

This paper investigates the spherical Fourier transform on harmonic Hadamard manifolds of hypergeometric type, providing explicit formulas and geometric characterizations for these special manifolds.

Contribution

It establishes the inversion, convolution, and Plancherel formulas for the Fourier transform on hypergeometric type manifolds using Gauss hypergeometric functions.

Findings

Explicit inversion formula for the spherical Fourier transform.

Convolution rule and Plancherel theorem derived for hypergeometric type.

Geometric characterization of hypergeometric type via volume density of geodesic spheres.

Abstract

The spherical Fourier transform on a harmonic Hadamard manifold $(X^n, g)$ of positive volume entropy is studied. If $(X^n, g)$ is of hypergeometric type, namely spherical functions of $X$ are represented by the Gauss hypergeometric functions, the inversion formula, the convolution rule together with the Plancherel theorem are shown by the representation of the spherical functions in terms of the Gauss hypergeometric functions. A geometric characterization of hypergeometric type is derived in terms of volume density of geodesic spheres. Geometric properties of $(X^n, g)$ are also discussed.