General Theory of Exponential Complex Metric Spaces: Signature Strata, Metric Transport, Embeddings, and Variable-Dimensional Geometry

Abstract

This monograph introduces exponential complex metric spaces as a coordinate-free framework for parameterizing nondegenerate symmetric bilinear forms and for organizing transport between their real and complex signature sectors. The construction is not the generalized metric of generalized geometry: it is defined on an ordinary vector bundle by composing a fixed nondegenerate reference metric with the exponential of a self-adjoint bundle endomorphism. The determinant identity $$[\det(g_0e^K)=\det(g_0)e^{\operatorname{tr}K}\]$$ makes nondegeneracy automatic. For real self-adjoint generators the signature is preserved. Positive-definite metrics admit a unique global relative logarithm, whereas indefinite metrics possess local exponential charts and finite piecewise-exponential descriptions but need not admit a single real self-adjoint logarithm between prescribed endpoints. After complexification every nondegenerate complex symmetric form admits an exponential representation relative to a fixed complex reference form, although that representation is not unique. Real signatures occur as distinguished phase slices of the complex space. Relative to a positive reference metric, every real nondegenerate metric also has a unique canonical polar–phase presentation: its positive absolute-value operator is the amplitude and its negative spectral projector carries phase \(\pi\). These data form a total exponential complex metric family whose realization map separates visible metric information from hidden presentation decoration. A moving positive carrier and a positive configuration-space inertia further define a Euclidean–inertial variational geometry. The normalized total scalar-curvature functional is pulled back to global self-adjoint generator coordinates; its stationary points remain exactly the Einstein metrics, while approximate metrics and nonstationary inertial trajectories are kept distinct. Energy identities, gauge directions, positivity-preserving Galerkin spaces, conditional compactness certificates, and sphere tests provide a computable framework without asserting unconditional Einstein existence or convergence. The remaining constructions treat metric transport, holonomy, pullback, embeddings, tangent–normal compression, stable transport, and variable-dimensional categories. Every global, local, and conditional conclusion is identified by its precise spectral, regularity, compactness, or gauge hypotheses. Keywords Exponential metric; complex metric; Euclidean action geometry; natural Lagrangian; Euler operator; configuration category; metric signature; matrix logarithm; tensor transport; metric connection; holonomy; isometric embedding; normal bundle; stable transport; variable-dimensional geometry; stratified space.