Invariant Projection Spaces: Integral Geometry and Canonical Metrics

We introduce the Principle of Invariant Projection and the associatedInvariant Projection Spaces (IPS), a unified framework for integralgeometry on homogeneous varieties. An IPS is a double fibration of incidenceendowed with an invariant probability measure, giving rise to a projectionoperator whose average extracts the G-invariant cohomology. The centrepieceis a fundamental identity ∫_X ω ∧ Π*(1) = ∫_G Π(ω) dμ, from which we deducethe classical Crofton and Schubert formulas. We introduce the notion ofIPS‑cohomological completeness and prove that, for generalised flag varietiesG/P, the Schubert cycles form a complete family; consequently the invariantprojection is injective on cohomology. This yields a rigorous characterisationof Kähler–Einstein metrics as those with constant invariant projection of theRicci form. An ergodic theorem for the iterated projection operator isestablished, leading to an IPS invariant; we conjecture its equality with theFutaki invariant and verify it for the Hirzebruch surface F_1. We constructIPS structures for Grassmannians, flag varieties, quadric hypersurfaces, andRiemannian spheres. The work culminates in the Mabuchi tower conjecture,linking the IPS iteration to the K-stability of Fano manifolds.