James J. Munden

By the Skew-Torsion Holonomy Theorem of Olmos and Reggiani (2012), the 15-sphere $S^{15}$ admits no non-trivial naturally reductive metric under any transitive Lie group, precluding geodesic integration via canonical Ambrose-Singer connections. Simultaneously, Nikonorov (2013) established that every $Spin(9)$-invariant metric on $S^{15}$ is a geodesic orbit (GO) space. We integrate the geodesic flow on this non-naturally reductive GO manifold via the Lie Subalgebra Closure Principle: whenever the slice generators of a multi-axial reductive decomposition close into a finite-rank reductive Lie subalgebra, the geodesic vector equations decouple into an algebraic eigenvalue problem. Using the 16-dimensional spin representation and Vol'per's decomposition $\mathfrak{spin}(9)=\mathfrak{spin}(7)\oplus {\mathfrak{p}}_1\oplus {\mathfrak{p}}_2$, we prove that the slice generators $X_1\in {\mathfrak{p}}_2$ and $Y_1\in {\mathfrak{p}}_1$ close into $\mathfrak{u}(2)\cong \mathfrak{su}(2)\oplus \mathfrak{u}(1)\subset \mathfrak{so}(16)$. The generator $M=\alpha X_1+\beta Y_1$ has the four-mode spectrum $\Omega \in \{|R\pm 3\alpha |,|R\pm \alpha |\}$ ($R=\sqrt{{\alpha }^2+{\beta }^2}$), yielding a closed-form 4-projector decomposition and Euler-Cartan factorization for $\exp (\alpha X_1+\beta Y_1)$. Under the normalized Ricci flow, we prove that the Bourguignon-Karcher squashed Einstein metric at $t=3/11$ ($\lambda =64$) is an unstable repeller separating sub-Riemannian fiber collapse from the round sphere basin. Finally, on the deformed Cayley plane $\mathbb{O}{P}^2=F_4/Spin(9)$, we prove universal CROSS never-Einstein rigidity, establish the 15-dimensional three-stratum Whitney stratification of the cut locus $Cut(q_0)$ at non-polar basepoints $q_0$, and prove Blaschke and Besse rigidity for all $\lambda >0$.

We study the one-parameter family of Riemannian metrics $g_{\lambda }$ induced by the Hamiltonian cotangent deformation $H_{\lambda }=H_0+\frac{\lambda }{2}\Vert J{\Vert }^2$ on simply connected space forms of constant curvature $\kappa \in \{1,0,-1\}$, where $J$ is the momentum map of the isotropy action fixing a base point. The master reciprocal identity $1/f_{\lambda }^2=1/s_{\kappa }^2+\lambda$ governing the warping function controls the geometry of the family and implies that radial geodesic acceleration is independent of $\lambda$. From this mechanism we derive three classes of results: (1) Curvature and rigidity: closed-form sectional curvatures, the sharp positivity trichotomy, pointwise and global Brendle-Schoen $1/4$-pinching windows on ${\mathbb{S}}^{n+1}$, never-Einstein rigidity for $\lambda \ne 0$, and the realization of Hamilton's steady cigar soliton on $({\mathbb{H}}^2,g_1)$ as the unique gradient Ricci soliton in the family. (2) Dynamics and stability: iso-kinematic decoupling into unperturbed radial motion and uniform precession $\Omega =\lambda L$, closed-form Clairaut integration, the transcendental inward conjugate point law with Euclidean inversion $r_0{R}^{\ast }=1/\lambda$, and stratified $A_3$ cusp caustics. (3) Integrability and spectral theory: maximal superintegrable rational pericenter invariants on ${\mathbb{R}}^{n+1}$, transverse Killing rigidity on curved space forms, Fuchsian Heun spectral reduction with modulus $\tau =1+\lambda /\kappa$, and essential spectrum gap collapse to $[0,\infty )$ on ${\mathbb{H}}^{n+1}$. Extensions to anisotropic sphere families and intrinsic deformations of compact rank-one symmetric spaces ($\mathbb{C}{P}^n,\mathbb{H}{P}^n,\mathbb{O}{P}^2$) are also established.