Banach PNDP-Manifolds: Fredholm Geometry, Symplectic Structures and the PNDP Maslov Index

We introduce and develop a Banach-manifold version of partially negative dimensional product manifolds (PNDP-manifolds). The construction is based on the observation that the virtual negative dimension of a PNDP fiber can be encoded analytically by a Fredholm index rather than by an ordinary finite-dimensional dimension. This leads naturally to a theory of Banach PNDP-manifolds in which the base is a Banach manifold, the fiber is modeled on a Banach derived-differential object, and the virtual dimension is defined by a Fredholm index. We establish basic structural results for Banach PNDP-manifolds, including a local normal form, a virtual-dimension formula, invariance under compact perturbations, and a criterion for the Einstein equation for warped Banach PNDP geometries under a finitedimensional reduction hypothesis. We then introduce strong and weak symplectic Banach PNDP-manifolds and prove that symplectic structures are compatible with the virtual decomposition when the positive and obstruction directions form a symplectic Fredholm pair. A central part of the paper concerns the Fredholm–Lagrangian Grassmannian. We define the PNDP Maslov index by reducing a path of Fredholm Lagrangian PNDP-subspaces to a finite-dimensional crossing problem. We prove homotopy invariance, additivity, concatenation, direct-sum additivity, naturality under symplectic isomorphisms, and invariance under admissible PNDP symplectic reduction. Several explicit examples are given, including constant negative-index bundles, a finite-dimensional PNDP model, a Hilbert-space loop with non-zero Maslov index, and a warped Banach PNDP model. The theory provides a common framework connecting PNDP geometry, Fredholm operators, Einstein-type warped products, symplectic Banach geometry, spectral flow, and Maslov index theory