  We perform global and local analysis of oscillatory and damped spherically
symmetric fundamental solutions for Helmholtz operators
$\big({-}\Delta\pm\beta^2\big)$ in $d$-dimensional, $R$-radius hyperbolic
${\mathbf H}_R^d$ and hyperspherical ${\mathbf S}_R^d$ geometry, which
represent Riemannian manifolds with positive constant and negative constant
sectional curvature respectively. In particular, we compute closed-form
expressions for fundamental solutions of $\big({-}\Delta \pm \beta^2\big)$ on
${\mathbf H}_R^d$, $\big({-}\Delta+\beta^2\big)$ on ${\mathbf S}_R^d$, and
present two candidate fundamental solutions for $\big({-}\Delta-\beta^2\big)$
on ${\mathbf S}_R^d$. Flat-space limits, with their corresponding asymptotic
representations, are used to restrict proportionality constants for these
fundamental solutions. In order to accomplish this, we summarize and derive new
large degree asymptotics for associated Legendre and Ferrers functions of the
first and second kind. Furthermore, we prove that our fundamental solutions on
the hyperboloid are unique due to their decay at infinity. To derive Gegenbauer
polynomial expansions of our fundamental solutions for Helmholtz operators on
hyperspheres and hyperboloids, we derive a collection of infinite series
addition theorems for Ferrers and associated Legendre functions which are
generalizations and extensions of the addition theorem for Gegenbauer
polynomials. Using these addition theorems, in geodesic polar coordinates for
dimensions greater than or equal to three, we compute Gegenbauer polynomial
expansions for these fundamental solutions, and azimuthal Fourier expansions in
two-dimensions.
