Speaker
Description
We present a unified geometric framework for hidden
symmetries in the broad class of Conformal St\"{a}ckel Spacetimes. While the
analytic separability of the Hamilton-Jacobi equation is well-established for
specific metrics, the geometric origin of the associated Conformal
Killing-St\"{a}ckel tensors has remained obscured by the limitations of standard
Killing-Yano theory, which often struggles with odd-dimensional defects and
higher-dimensional topological bases. We resolve this limitation by inverting
the standard logical hierarchy: we establish that the total umbilicity of
orthogonal coordinate distributions is not merely a consequence of separability,
but its fundamental, conformally invariant geometric generator. Proving that
total umbilicity is the necessary and sufficient condition for conformal
St\"{a}ckel separability, we introduce the Symplectic Extension Map. This map
constructs non-degenerate Symplectic Blocks by organically pairing defective
umbilical foliations with conjugate isometries, naturally preserving dimensional
consistency via the Cartan subalgebra of the topological sectors. By utilizing
the essential St\"{a}ckel coordinates as spectral weights (degenerate
eigenvalues) for these blocks, we construct the Generalised Killing-Yano
Polyform: an inhomogeneous element of the exterior algebra defined by the
exterior exponential of the block symplectic forms. We prove that the metric
square of this polyform strictly generates the complete Conformal Killing tower.
Consequently, this framework reveals that the standard Principal Conformal
Killing-Yano Tensor (PCKYT) of the Kerr-NUT-AdS family is not an isolated metric
coincidence, but the lowest-dimensional special case of a universal algebraic
structure, providing a rigorous geometric description for hidden symmetries
across arbitrary dimensions and topologically non-trivial backgrounds.